Calculus Examples

$f (x) = \frac{{(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}}}{- 3 x^{- 3} - 2 x + 6}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}}$ and $g (x) = - 3 x^{- 3} - 2 x + 6$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{d}{d x} [{(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}}] - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{4}{3}}$ and $g (x) = 8 x^{- 4} + 4 x + 10$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $8 x^{- 4} + 4 x + 10$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{d}{d u} [u^{\frac{4}{3}}] \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{4}{3}$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{4}{3} u^{\frac{4}{3} - 1} \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 2.3

Replace all occurrences of $u$ with $8 x^{- 4} + 4 x + 10$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{4}{3} {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3} - 1} \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{4}{3} {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 4

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{4}{3} {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{4}{3} {(8 x^{- 4} + 4 x + 10)}^{\frac{4 - 1 \cdot 3}{3}} \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{4}{3} {(8 x^{- 4} + 4 x + 10)}^{\frac{4 - 3}{3}} \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 6.2

Subtract $3$ from $4$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{4}{3} {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}} \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 7

Combine $\frac{4}{3}$ and ${(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) (\frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} \frac{d}{d x} [8 x^{- 4} + 4 x + 10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 8

By the Sum Rule, the derivative of $8 x^{- 4} + 4 x + 10$ with respect to $x$ is $\frac{d}{d x} [8 x^{- 4}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [10]$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (\frac{d}{d x} [8 x^{- 4}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 9

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{- 4}$ with respect to $x$ is $8 \frac{d}{d x} [x^{- 4}]$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (8 \frac{d}{d x} [x^{- 4}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 4$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (8 (- 4 x^{- 5}) + \frac{d}{d x} [4 x] + \frac{d}{d x} [10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 11

Multiply $- 4$ by $8$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + \frac{d}{d x} [4 x] + \frac{d}{d x} [10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 12

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4 \frac{d}{d x} [x] + \frac{d}{d x} [10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4 \cdot 1 + \frac{d}{d x} [10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 14

Multiply $4$ by $1$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4 + \frac{d}{d x} [10]) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 15

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4 + 0) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 16

Add $- 32 x^{- 5} + 4$ and $0$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} \frac{d}{d x} [- 3 x^{- 3} - 2 x + 6]}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 17

By the Sum Rule, the derivative of $- 3 x^{- 3} - 2 x + 6$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{- 3}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [6]$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (\frac{d}{d x} [- 3 x^{- 3}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [6])}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 18

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{- 3}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{- 3}]$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (- 3 \frac{d}{d x} [x^{- 3}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [6])}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 19

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (- 3 (- 3 x^{- 4}) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [6])}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 20

Multiply $- 3$ by $- 3$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [6])}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 21

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2 \frac{d}{d x} [x] + \frac{d}{d x} [6])}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 22

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2 \cdot 1 + \frac{d}{d x} [6])}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 23

Multiply $- 2$ by $1$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2 + \frac{d}{d x} [6])}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 24

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2 + 0)}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 25

Add $9 x^{- 4} - 2$ and $0$ .

$\frac{(- 3 x^{- 3} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2)}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 26

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(8 x^{- 4} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2)}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 27

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 x^{- 5} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2)}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 28

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 x^{- 4} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2)}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 29

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 x^{- 4} - 2)}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 30

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 x^{- 3} - 2 x + 6)}^{2}}$

Step 31

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

Step 32

Simplify.

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Step 32.1

Simplify the numerator.

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Step 32.1.1

Simplify the numerator.

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Step 32.1.1.1

Combine $8$ and $\frac{1}{x^{4}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.2

To write $4 x$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{8}{x^{4}} + \frac{4 x \cdot x^{4}}{x^{4}} + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.3

Combine the numerators over the common denominator.

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{8 + 4 x \cdot x^{4}}{x^{4}} + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.4

Simplify the numerator.

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Step 32.1.1.4.1

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 32.1.1.4.1.1

Move $x^{4}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{8 + 4 (x^{4} x)}{x^{4}} + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.4.1.2

Multiply $x^{4}$ by $x$ .

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Step 32.1.1.4.1.2.1

Raise $x$ to the power of $1$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{8 + 4 (x^{4} x^{1})}{x^{4}} + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{8 + 4 x^{4 + 1}}{x^{4}} + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.4.1.3

Add $4$ and $1$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{8 + 4 x^{5}}{x^{4}} + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.4.2

Factor $4$ out of $8 + 4 x^{5}$ .

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Step 32.1.1.4.2.1

Factor $4$ out of $8$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{4 \cdot 2 + 4 x^{5}}{x^{4}} + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.4.2.2

Factor $4$ out of $4 \cdot 2 + 4 x^{5}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{4 (2 + x^{5})}{x^{4}} + 10)}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.5

To write $10$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{4 (2 + x^{5})}{x^{4}} + \frac{10 x^{4}}{x^{4}})}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.6

Combine the numerators over the common denominator.

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{4 (2 + x^{5}) + 10 x^{4}}{x^{4}})}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.7

Simplify the numerator.

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Step 32.1.1.7.1

Factor $2$ out of $4 (2 + x^{5}) + 10 x^{4}$ .

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Step 32.1.1.7.1.1

Factor $2$ out of $4 (2 + x^{5})$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{2 (2 (2 + x^{5})) + 10 x^{4}}{x^{4}})}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.7.1.2

Factor $2$ out of $10 x^{4}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{2 (2 (2 + x^{5})) + 2 (5 x^{4})}{x^{4}})}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.7.1.3

Factor $2$ out of $2 (2 (2 + x^{5})) + 2 (5 x^{4})$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{2 (2 (2 + x^{5}) + 5 x^{4})}{x^{4}})}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.7.2

Apply the distributive property.

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{2 (2 \cdot 2 + 2 x^{5} + 5 x^{4})}{x^{4}})}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.7.3

Multiply $2$ by $2$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{2 (4 + 2 x^{5} + 5 x^{4})}{x^{4}})}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.7.4

Reorder terms.

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 {(\frac{2 (2 x^{5} + 5 x^{4} + 4)}{x^{4}})}^{\frac{1}{3}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 32.1.1.8.1

Apply the product rule to $\frac{2 (2 x^{5} + 5 x^{4} + 4)}{x^{4}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 \frac{{(2 (2 x^{5} + 5 x^{4} + 4))}^{\frac{1}{3}}}{{(x^{4})}^{\frac{1}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.8.2

Apply the product rule to $2 (2 x^{5} + 5 x^{4} + 4)$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 \frac{2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{{(x^{4})}^{\frac{1}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.9

Multiply the exponents in ${(x^{4})}^{\frac{1}{3}}$ .

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Step 32.1.1.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 \frac{2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{4 (\frac{1}{3})}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.1.9.2

Combine $4$ and $\frac{1}{3}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{4 \frac{2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.2

Combine $4$ and $\frac{2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{\frac{4 (2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}})}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.3

Simplify the numerator.

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Step 32.1.3.1

Rewrite $4$ as $2^{2}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{\frac{2^{2} \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{\frac{2^{2 + \frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.3.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{\frac{2^{2 \cdot \frac{3}{3} + \frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.3.4

Combine $2$ and $\frac{3}{3}$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{\frac{2^{\frac{2 \cdot 3}{3} + \frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.3.5

Combine the numerators over the common denominator.

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{\frac{2^{\frac{2 \cdot 3 + 1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.3.6

Simplify the numerator.

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Step 32.1.3.6.1

Multiply $2$ by $3$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{\frac{2^{\frac{6 + 1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.3.6.2

Add $6$ and $1$ .

$\frac{(- 3 \frac{1}{x^{3}} - 2 x + 6) \frac{\frac{2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.4

Simplify each term.

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Step 32.1.4.1

Combine $- 3$ and $\frac{1}{x^{3}}$ .

$\frac{(\frac{- 3}{x^{3}} - 2 x + 6) \frac{\frac{2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.4.2

Move the negative in front of the fraction.

$\frac{(- \frac{3}{x^{3}} - 2 x + 6) \frac{\frac{2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}}}{3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{(- \frac{3}{x^{3}} - 2 x + 6) (\frac{2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}}} \cdot \frac{1}{3}) (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.6

Combine.

$\frac{(- \frac{3}{x^{3}} - 2 x + 6) \frac{2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{\frac{4}{3}} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.7

Multiply $2^{\frac{7}{3}}$ by $1$ .

$\frac{(- \frac{3}{x^{3}} - 2 x + 6) \frac{2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{4}{3}} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.8

Move $3$ to the left of $x^{\frac{4}{3}}$ .

$\frac{(- \frac{3}{x^{3}} - 2 x + 6) \frac{2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 \cdot x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.9

Multiply $- \frac{3}{x^{3}} - 2 x + 6$ by $\frac{2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}}$ .

$\frac{\frac{(- \frac{3}{x^{3}} - 2 x + 6) (2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}})}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10

Simplify the numerator.

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Step 32.1.10.1

Reorder terms.

$\frac{\frac{(- 2 x - \frac{3}{x^{3}} + 6) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.2

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{\frac{(- 2 x \cdot \frac{x^{3}}{x^{3}} - \frac{3}{x^{3}} + 6) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.3

Combine $- 2 x$ and $\frac{x^{3}}{x^{3}}$ .

$\frac{\frac{(\frac{- 2 x \cdot x^{3}}{x^{3}} - \frac{3}{x^{3}} + 6) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.4

Combine the numerators over the common denominator.

$\frac{\frac{(\frac{- 2 x \cdot x^{3} - 3}{x^{3}} + 6) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.5

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 32.1.10.5.1

Move $x^{3}$ .

$\frac{\frac{(\frac{- 2 (x^{3} x) - 3}{x^{3}} + 6) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.5.2

Multiply $x^{3}$ by $x$ .

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Step 32.1.10.5.2.1

Raise $x$ to the power of $1$ .

$\frac{\frac{(\frac{- 2 (x^{3} x^{1}) - 3}{x^{3}} + 6) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{(\frac{- 2 x^{3 + 1} - 3}{x^{3}} + 6) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.5.3

Add $3$ and $1$ .

$\frac{\frac{(\frac{- 2 x^{4} - 3}{x^{3}} + 6) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.6

To write $6$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{\frac{(\frac{- 2 x^{4} - 3}{x^{3}} + \frac{6 x^{3}}{x^{3}}) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.7

Combine the numerators over the common denominator.

$\frac{\frac{\frac{- 2 x^{4} - 3 + 6 x^{3}}{x^{3}} \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.8

Reorder terms.

$\frac{\frac{\frac{- 2 x^{4} + 6 x^{3} - 3}{x^{3}} \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.9

Combine exponents.

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Step 32.1.10.9.1

Combine $\frac{- 2 x^{4} + 6 x^{3} - 3}{x^{3}}$ and $2^{\frac{7}{3}}$ .

$\frac{\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}}}{x^{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.10.9.2

Combine $\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}}}{x^{3}}$ and ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}$ .

$\frac{\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{3}}}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.11

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{3}} \cdot \frac{1}{3 x^{\frac{4}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.12

Combine.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{3} (3 x^{\frac{4}{3}})} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.13

Multiply $x^{3}$ by $x^{\frac{4}{3}}$ by adding the exponents.

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Step 32.1.13.1

Move $x^{\frac{4}{3}}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{\frac{4}{3}} x^{3} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.13.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{\frac{4}{3} + 3} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.13.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{\frac{4}{3} + 3 \cdot \frac{3}{3}} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.13.4

Combine $3$ and $\frac{3}{3}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{\frac{4}{3} + \frac{3 \cdot 3}{3}} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.13.5

Combine the numerators over the common denominator.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{\frac{4 + 3 \cdot 3}{3}} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.13.6

Simplify the numerator.

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Step 32.1.13.6.1

Multiply $3$ by $3$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{\frac{4 + 9}{3}} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.13.6.2

Add $4$ and $9$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 1}{x^{\frac{13}{3}} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.14

Multiply $(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}$ by $1$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{x^{\frac{13}{3}} \cdot 3} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.15

Move $3$ to the left of $x^{\frac{13}{3}}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 \cdot x^{\frac{13}{3}}} (- 32 \frac{1}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.16

Simplify each term.

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Step 32.1.16.1

Combine $- 32$ and $\frac{1}{x^{5}}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{13}{3}}} (\frac{- 32}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.16.2

Move the negative in front of the fraction.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{13}{3}}} (- \frac{32}{x^{5}} + 4) - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.17

Multiply $\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{13}{3}}}$ by $- \frac{32}{x^{5}} + 4$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{32}{x^{5}} + 4)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18

Simplify the numerator.

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Step 32.1.18.1

Factor $4$ out of $- \frac{32}{x^{5}} + 4$ .

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Step 32.1.18.1.1

Factor $4$ out of $- \frac{32}{x^{5}}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (4 (- \frac{8}{x^{5}}) + 4)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.1.2

Factor $4$ out of $4$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (4 (- \frac{8}{x^{5}}) + 4 (1))}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.1.3

Factor $4$ out of $4 (- \frac{8}{x^{5}}) + 4 (1)$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (4 (- \frac{8}{x^{5}} + 1))}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 4 (- \frac{8}{x^{5}} + 1)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.2

Combine exponents.

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Step 32.1.18.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot (2^{2} \cdot 2^{\frac{7}{3}}) {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{8}{x^{5}} + 1)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{2 + \frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{8}{x^{5}} + 1)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{2 \cdot \frac{3}{3} + \frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{8}{x^{5}} + 1)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.2.4

Combine $2$ and $\frac{3}{3}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{2 \cdot 3}{3} + \frac{7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{8}{x^{5}} + 1)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.2.5

Combine the numerators over the common denominator.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{2 \cdot 3 + 7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{8}{x^{5}} + 1)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.2.6

Simplify the numerator.

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Step 32.1.18.2.6.1

Multiply $2$ by $3$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{6 + 7}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{8}{x^{5}} + 1)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.2.6.2

Add $6$ and $7$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{13}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{8}{x^{5}} + 1)}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.3

Write $1$ as a fraction with a common denominator.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{13}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- \frac{8}{x^{5}} + \frac{x^{5}}{x^{5}})}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.4

Combine the numerators over the common denominator.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot 2^{\frac{13}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \frac{- 8 + x^{5}}{x^{5}}}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.5

Combine exponents.

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Step 32.1.18.5.1

Combine $\frac{- 8 + x^{5}}{x^{5}}$ and $2^{\frac{13}{3}}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot \frac{(- 8 + x^{5}) \cdot 2^{\frac{13}{3}}}{x^{5}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.5.2

Combine ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}$ and $\frac{(- 8 + x^{5}) \cdot 2^{\frac{13}{3}}}{x^{5}}$ .

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot \frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} ((- 8 + x^{5}) \cdot 2^{\frac{13}{3}})}{x^{5}}}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.18.6

Remove unnecessary parentheses.

$\frac{\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot \frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}}}{x^{5}}}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.19

Factor $\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}}}{x^{5}}$ out of $\frac{(- 2 x^{4} + 6 x^{3} - 3) \cdot \frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}}}{x^{5}}}{3 x^{\frac{13}{3}}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}}}{x^{5}} \cdot \frac{- 2 x^{4} + 6 x^{3} - 3}{3 x^{\frac{13}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.20

Multiply $\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}}}{x^{5}} \cdot \frac{- 2 x^{4} + 6 x^{3} - 3}{3 x^{\frac{13}{3}}}$ .

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Step 32.1.20.1

Multiply $\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}}}{x^{5}}$ by $\frac{- 2 x^{4} + 6 x^{3} - 3}{3 x^{\frac{13}{3}}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{x^{5} (3 x^{\frac{13}{3}})} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.20.2

Multiply $x^{5}$ by $x^{\frac{13}{3}}$ by adding the exponents.

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Step 32.1.20.2.1

Move $x^{\frac{13}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{x^{\frac{13}{3}} x^{5} \cdot 3} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.20.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{x^{\frac{13}{3} + 5} \cdot 3} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.20.2.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{x^{\frac{13}{3} + 5 \cdot \frac{3}{3}} \cdot 3} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.20.2.4

Combine $5$ and $\frac{3}{3}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{x^{\frac{13}{3} + \frac{5 \cdot 3}{3}} \cdot 3} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.20.2.5

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{x^{\frac{13 + 5 \cdot 3}{3}} \cdot 3} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.20.2.6

Simplify the numerator.

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Step 32.1.20.2.6.1

Multiply $5$ by $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{x^{\frac{13 + 15}{3}} \cdot 3} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.20.2.6.2

Add $13$ and $15$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{x^{\frac{28}{3}} \cdot 3} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.21

Move $3$ to the left of $x^{\frac{28}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 \cdot x^{\frac{28}{3}}} - {(8 \frac{1}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.22

Combine $8$ and $\frac{1}{x^{4}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (9 \frac{1}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.23

Combine $9$ and $\frac{1}{x^{4}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} (\frac{9}{x^{4}} - 2)}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.24

Apply the distributive property.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} \frac{9}{x^{4}} - {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} \cdot - 2}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.25

Combine $\frac{9}{x^{4}}$ and ${(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{x^{4}} - {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} \cdot - 2}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.26

Multiply $- 2$ by $- 1$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27

Simplify each term.

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Step 32.1.27.1

Simplify the numerator.

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Step 32.1.27.1.1

To write $4 x$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{8}{x^{4}} + \frac{4 x \cdot x^{4}}{x^{4}} + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.2

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{8 + 4 x \cdot x^{4}}{x^{4}} + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.3

Simplify the numerator.

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Step 32.1.27.1.3.1

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 32.1.27.1.3.1.1

Move $x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{8 + 4 (x^{4} x)}{x^{4}} + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.3.1.2

Multiply $x^{4}$ by $x$ .

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Step 32.1.27.1.3.1.2.1

Raise $x$ to the power of $1$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{8 + 4 (x^{4} x^{1})}{x^{4}} + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{8 + 4 x^{4 + 1}}{x^{4}} + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.3.1.3

Add $4$ and $1$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{8 + 4 x^{5}}{x^{4}} + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.3.2

Factor $4$ out of $8 + 4 x^{5}$ .

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Step 32.1.27.1.3.2.1

Factor $4$ out of $8$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{4 \cdot 2 + 4 x^{5}}{x^{4}} + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.3.2.2

Factor $4$ out of $4 \cdot 2 + 4 x^{5}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{4 (2 + x^{5})}{x^{4}} + 10)}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.4

To write $10$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{4 (2 + x^{5})}{x^{4}} + \frac{10 x^{4}}{x^{4}})}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.5

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{4 (2 + x^{5}) + 10 x^{4}}{x^{4}})}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.6

Simplify the numerator.

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Step 32.1.27.1.6.1

Factor $2$ out of $4 (2 + x^{5}) + 10 x^{4}$ .

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Step 32.1.27.1.6.1.1

Factor $2$ out of $4 (2 + x^{5})$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{2 (2 (2 + x^{5})) + 10 x^{4}}{x^{4}})}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.6.1.2

Factor $2$ out of $10 x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{2 (2 (2 + x^{5})) + 2 (5 x^{4})}{x^{4}})}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.6.1.3

Factor $2$ out of $2 (2 (2 + x^{5})) + 2 (5 x^{4})$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{2 (2 (2 + x^{5}) + 5 x^{4})}{x^{4}})}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.6.2

Apply the distributive property.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{2 (2 \cdot 2 + 2 x^{5} + 5 x^{4})}{x^{4}})}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.6.3

Multiply $2$ by $2$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{2 (4 + 2 x^{5} + 5 x^{4})}{x^{4}})}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.6.4

Reorder terms.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 {(\frac{2 (2 x^{5} + 5 x^{4} + 4)}{x^{4}})}^{\frac{4}{3}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 32.1.27.1.7.1

Apply the product rule to $\frac{2 (2 x^{5} + 5 x^{4} + 4)}{x^{4}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \frac{{(2 (2 x^{5} + 5 x^{4} + 4))}^{\frac{4}{3}}}{{(x^{4})}^{\frac{4}{3}}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.7.2

Apply the product rule to $2 (2 x^{5} + 5 x^{4} + 4)$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{{(x^{4})}^{\frac{4}{3}}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.8

Multiply the exponents in ${(x^{4})}^{\frac{4}{3}}$ .

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Step 32.1.27.1.8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{4 (\frac{4}{3})}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.8.2

Multiply $4 (\frac{4}{3})$ .

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Step 32.1.27.1.8.2.1

Combine $4$ and $\frac{4}{3}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{4 \cdot 4}{3}}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.1.8.2.2

Multiply $4$ by $4$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{16}{3}}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.2

Combine $9$ and $\frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{16}{3}}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{\frac{9 (2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}})}{x^{\frac{16}{3}}}}{x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - (\frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{16}{3}}} \cdot \frac{1}{x^{4}}) + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.4

Combine.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} \cdot 1}{x^{\frac{16}{3}} x^{4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.5

Multiply $x^{\frac{16}{3}}$ by $x^{4}$ by adding the exponents.

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Step 32.1.27.5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} \cdot 1}{x^{\frac{16}{3} + 4}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.5.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} \cdot 1}{x^{\frac{16}{3} + 4 \cdot \frac{3}{3}}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.5.3

Combine $4$ and $\frac{3}{3}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} \cdot 1}{x^{\frac{16}{3} + \frac{4 \cdot 3}{3}}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.5.4

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} \cdot 1}{x^{\frac{16 + 4 \cdot 3}{3}}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.5.5

Simplify the numerator.

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Step 32.1.27.5.5.1

Multiply $4$ by $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} \cdot 1}{x^{\frac{16 + 12}{3}}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.5.5.2

Add $16$ and $12$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} \cdot 1}{x^{\frac{28}{3}}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.27.6

Multiply $9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ by $1$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{28}{3}}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.28

To write $2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{28}{3}}}{x^{\frac{28}{3}}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} - \frac{9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{28}{3}}} + \frac{2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} x^{\frac{28}{3}}}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.29

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{- 9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} x^{\frac{28}{3}}}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30

Simplify the numerator.

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Step 32.1.30.1

Factor $2$ out of $- 9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} x^{\frac{28}{3}}$ .

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Step 32.1.30.1.1

Factor $2$ out of $- 9 \cdot 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}) + 2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} x^{\frac{28}{3}}}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.1.2

Factor $2$ out of $2 {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} x^{\frac{28}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}) + 2 ({(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.1.3

Factor $2$ out of $2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}) + 2 ({(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{8}{x^{4}} + 4 x + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2

Simplify each term.

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Step 32.1.30.2.1

To write $4 x$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{8}{x^{4}} + \frac{4 x \cdot x^{4}}{x^{4}} + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.2

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{8 + 4 x \cdot x^{4}}{x^{4}} + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.3

Simplify the numerator.

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Step 32.1.30.2.3.1

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 32.1.30.2.3.1.1

Move $x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{8 + 4 (x^{4} x)}{x^{4}} + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.3.1.2

Multiply $x^{4}$ by $x$ .

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Step 32.1.30.2.3.1.2.1

Raise $x$ to the power of $1$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{8 + 4 (x^{4} x^{1})}{x^{4}} + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{8 + 4 x^{4 + 1}}{x^{4}} + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.3.1.3

Add $4$ and $1$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{8 + 4 x^{5}}{x^{4}} + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.3.2

Factor $4$ out of $8 + 4 x^{5}$ .

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Step 32.1.30.2.3.2.1

Factor $4$ out of $8$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{4 \cdot 2 + 4 x^{5}}{x^{4}} + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.3.2.2

Factor $4$ out of $4 \cdot 2 + 4 x^{5}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{4 (2 + x^{5})}{x^{4}} + 10)}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.4

To write $10$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{4 (2 + x^{5})}{x^{4}} + \frac{10 x^{4}}{x^{4}})}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.5

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{4 (2 + x^{5}) + 10 x^{4}}{x^{4}})}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.6

Simplify the numerator.

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Step 32.1.30.2.6.1

Factor $2$ out of $4 (2 + x^{5}) + 10 x^{4}$ .

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Step 32.1.30.2.6.1.1

Factor $2$ out of $4 (2 + x^{5})$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{2 (2 (2 + x^{5})) + 10 x^{4}}{x^{4}})}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.6.1.2

Factor $2$ out of $10 x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{2 (2 (2 + x^{5})) + 2 (5 x^{4})}{x^{4}})}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.6.1.3

Factor $2$ out of $2 (2 (2 + x^{5})) + 2 (5 x^{4})$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{2 (2 (2 + x^{5}) + 5 x^{4})}{x^{4}})}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.6.2

Apply the distributive property.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{2 (2 \cdot 2 + 2 x^{5} + 5 x^{4})}{x^{4}})}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.6.3

Multiply $2$ by $2$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{2 (4 + 2 x^{5} + 5 x^{4})}{x^{4}})}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.6.4

Reorder terms.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + {(\frac{2 (2 x^{5} + 5 x^{4} + 4)}{x^{4}})}^{\frac{4}{3}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 32.1.30.2.7.1

Apply the product rule to $\frac{2 (2 x^{5} + 5 x^{4} + 4)}{x^{4}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + \frac{{(2 (2 x^{5} + 5 x^{4} + 4))}^{\frac{4}{3}}}{{(x^{4})}^{\frac{4}{3}}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.7.2

Apply the product rule to $2 (2 x^{5} + 5 x^{4} + 4)$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{{(x^{4})}^{\frac{4}{3}}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.8

Multiply the exponents in ${(x^{4})}^{\frac{4}{3}}$ .

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Step 32.1.30.2.8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{4 (\frac{4}{3})}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.8.2

Multiply $4 (\frac{4}{3})$ .

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Step 32.1.30.2.8.2.1

Combine $4$ and $\frac{4}{3}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{4 \cdot 4}{3}}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.8.2.2

Multiply $4$ by $4$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{16}{3}}} x^{\frac{28}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.9

Cancel the common factor of $x^{\frac{16}{3}}$ .

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Step 32.1.30.2.9.1

Factor $x^{\frac{16}{3}}$ out of $x^{\frac{28}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{16}{3}}} (x^{\frac{16}{3}} x^{\frac{12}{3}}))}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.9.2

Cancel the common factor.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}}{x^{\frac{16}{3}}} (x^{\frac{16}{3}} x^{\frac{12}{3}}))}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.9.3

Rewrite the expression.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} x^{\frac{12}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.2.10

Divide $12$ by $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} x^{4})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.3

Factor $2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ out of $- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} x^{4}$ .

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Step 32.1.30.3.1

Move ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} + 2^{\frac{4}{3}} x^{4} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.3.2

Factor $2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ out of $- 9 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9) + 2^{\frac{4}{3}} x^{4} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.3.3

Factor $2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ out of $2^{\frac{4}{3}} x^{4} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9) + 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (2^{\frac{3}{3}} x^{4}))}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.3.4

Factor $2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ out of $2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9) + 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (2^{\frac{3}{3}} x^{4})$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2^{\frac{3}{3}} x^{4}))}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.4

Divide $3$ by $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 (2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2^{1} x^{4}))}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.5

Evaluate the exponent.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2 \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.6

Multiply $2$ by $2^{\frac{1}{3}}$ by adding the exponents.

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Step 32.1.30.6.1

Multiply $2$ by $2^{\frac{1}{3}}$ .

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Step 32.1.30.6.1.1

Raise $2$ to the power of $1$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2^{1} \cdot 2^{\frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2^{1 + \frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.6.2

Write $1$ as a fraction with a common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2^{\frac{3}{3} + \frac{1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.6.3

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2^{\frac{3 + 1}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.30.6.4

Add $3$ and $1$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4})}{x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.31

To write $\frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4})}{x^{\frac{28}{3}}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

Step 32.1.32

Write each expression with a common denominator of $3 x^{\frac{28}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 32.1.32.1

Multiply $\frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4})}{x^{\frac{28}{3}}}$ by $\frac{3}{3}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4}) \cdot 3}{x^{\frac{28}{3}} \cdot 3}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.32.2

Reorder the factors of $x^{\frac{28}{3}} \cdot 3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3)}{3 x^{\frac{28}{3}}} + \frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4}) \cdot 3}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.33

Combine the numerators over the common denominator.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4}) \cdot 3}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34

Rewrite $\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4}) \cdot 3}{3 x^{\frac{28}{3}}}$ in a factored form.

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Step 32.1.34.1

Factor ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}}$ out of ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 8 + x^{5}) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4}) \cdot 3$ .

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Step 32.1.34.1.1

Reorder the expression.

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Step 32.1.34.1.1.1

Reorder $- 8$ and $x^{5}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (x^{5} - 8) \cdot 2^{\frac{13}{3}} (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4}) \cdot 3}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.1.1.2

Move $x^{5} - 8$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{13}{3}} (x^{5} - 8) (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4}) \cdot 3}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.1.1.3

Reorder ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}}$ and $2^{\frac{13}{3}}$ .

$\frac{\frac{2^{\frac{13}{3}} \cdot {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (x^{5} - 8) (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (- 9 + 2 x^{4}) \cdot 3}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.1.1.4

Reorder $- 9$ and $2 x^{4}$ .

$\frac{\frac{2^{\frac{13}{3}} \cdot {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (x^{5} - 8) (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (2 x^{4} - 9) \cdot 3}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.1.1.5

Move $2 x^{4} - 9$ .

$\frac{\frac{2^{\frac{13}{3}} \cdot {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (x^{5} - 8) (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} \cdot 3 (2 x^{4} - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.1.1.6

Move ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}}$ .

$\frac{\frac{2^{\frac{13}{3}} \cdot {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (x^{5} - 8) (- 2 x^{4} + 6 x^{3} - 3) + 2^{\frac{4}{3}} \cdot 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (2 x^{4} - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.1.2

Factor ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}}$ out of $2^{\frac{13}{3}} \cdot {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (x^{5} - 8) (- 2 x^{4} + 6 x^{3} - 3)$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} ((2^{\frac{9}{3}} (x^{5} - 8)) (- 2 x^{4} + 6 x^{3} - 3)) + 2^{\frac{4}{3}} \cdot 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (2 x^{4} - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.1.3

Factor ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}}$ out of $2^{\frac{4}{3}} \cdot 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{4}{3}} (2 x^{4} - 9)$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} ((2^{\frac{9}{3}} (x^{5} - 8)) (- 2 x^{4} + 6 x^{3} - 3)) + {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.1.4

Factor ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}}$ out of ${(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} ((2^{\frac{9}{3}} (x^{5} - 8)) (- 2 x^{4} + 6 x^{3} - 3)) + {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} ((2^{\frac{9}{3}} (x^{5} - 8)) (- 2 x^{4} + 6 x^{3} - 3) + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.2

Divide $9$ by $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (2^{3} (x^{5} - 8) (- 2 x^{4} + 6 x^{3} - 3) + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.3

Raise $2$ to the power of $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (8 (x^{5} - 8) (- 2 x^{4} + 6 x^{3} - 3) + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.4

Apply the distributive property.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} ((8 x^{5} + 8 \cdot - 8) (- 2 x^{4} + 6 x^{3} - 3) + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.5

Multiply $8$ by $- 8$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} ((8 x^{5} - 64) (- 2 x^{4} + 6 x^{3} - 3) + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.6

Expand $(8 x^{5} - 64) (- 2 x^{4} + 6 x^{3} - 3)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (8 x^{5} (- 2 x^{4}) + 8 x^{5} (6 x^{3}) + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7

Simplify each term.

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Step 32.1.34.7.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (8 \cdot - 2 x^{5} x^{4} + 8 x^{5} (6 x^{3}) + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.2

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 32.1.34.7.2.1

Move $x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (8 \cdot - 2 (x^{4} x^{5}) + 8 x^{5} (6 x^{3}) + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (8 \cdot - 2 x^{4 + 5} + 8 x^{5} (6 x^{3}) + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.2.3

Add $4$ and $5$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (8 \cdot - 2 x^{9} + 8 x^{5} (6 x^{3}) + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.3

Multiply $8$ by $- 2$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 8 x^{5} (6 x^{3}) + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.4

Rewrite using the commutative property of multiplication.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 8 \cdot 6 x^{5} x^{3} + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.5

Multiply $x^{5}$ by $x^{3}$ by adding the exponents.

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Step 32.1.34.7.5.1

Move $x^{3}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 8 \cdot 6 (x^{3} x^{5}) + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 8 \cdot 6 x^{3 + 5} + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.5.3

Add $3$ and $5$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 8 \cdot 6 x^{8} + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.6

Multiply $8$ by $6$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} + 8 x^{5} \cdot - 3 - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.7

Multiply $- 3$ by $8$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} - 64 (- 2 x^{4}) - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.8

Multiply $- 2$ by $- 64$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 64 (6 x^{3}) - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.9

Multiply $6$ by $- 64$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} - 64 \cdot - 3 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.7.10

Multiply $- 64$ by $- 3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{\frac{3}{3}} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.8

Divide $3$ by $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 3 {(2 x^{5} + 5 x^{4} + 4)}^{1} (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.9

Simplify.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 3 (2 x^{5} + 5 x^{4} + 4) (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.10

Apply the distributive property.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + (3 (2 x^{5}) + 3 (5 x^{4}) + 3 \cdot 4) (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.11

Simplify.

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Step 32.1.34.11.1

Multiply $2$ by $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + (6 x^{5} + 3 (5 x^{4}) + 3 \cdot 4) (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.11.2

Multiply $5$ by $3$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + (6 x^{5} + 15 x^{4} + 3 \cdot 4) (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.11.3

Multiply $3$ by $4$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + (6 x^{5} + 15 x^{4} + 12) (2 x^{4} - 9))}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.12

Expand $(6 x^{5} + 15 x^{4} + 12) (2 x^{4} - 9)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 6 x^{5} (2 x^{4}) + 6 x^{5} \cdot - 9 + 15 x^{4} (2 x^{4}) + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13

Simplify each term.

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Step 32.1.34.13.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 6 \cdot 2 x^{5} x^{4} + 6 x^{5} \cdot - 9 + 15 x^{4} (2 x^{4}) + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.2

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 32.1.34.13.2.1

Move $x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 6 \cdot 2 (x^{4} x^{5}) + 6 x^{5} \cdot - 9 + 15 x^{4} (2 x^{4}) + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 6 \cdot 2 x^{4 + 5} + 6 x^{5} \cdot - 9 + 15 x^{4} (2 x^{4}) + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.2.3

Add $4$ and $5$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 6 \cdot 2 x^{9} + 6 x^{5} \cdot - 9 + 15 x^{4} (2 x^{4}) + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.3

Multiply $6$ by $2$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} + 6 x^{5} \cdot - 9 + 15 x^{4} (2 x^{4}) + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.4

Multiply $- 9$ by $6$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 15 x^{4} (2 x^{4}) + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.5

Rewrite using the commutative property of multiplication.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 15 \cdot 2 x^{4} x^{4} + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.6

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

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Step 32.1.34.13.6.1

Move $x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 15 \cdot 2 (x^{4} x^{4}) + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 15 \cdot 2 x^{4 + 4} + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.6.3

Add $4$ and $4$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 15 \cdot 2 x^{8} + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.7

Multiply $15$ by $2$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 30 x^{8} + 15 x^{4} \cdot - 9 + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.8

Multiply $- 9$ by $15$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 30 x^{8} - 135 x^{4} + 12 (2 x^{4}) + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.9

Multiply $2$ by $12$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 30 x^{8} - 135 x^{4} + 24 x^{4} + 12 \cdot - 9)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.13.10

Multiply $12$ by $- 9$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 30 x^{8} - 135 x^{4} + 24 x^{4} - 108)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.14

Add $- 135 x^{4}$ and $24 x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 16 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 + 12 x^{9} - 54 x^{5} + 30 x^{8} - 111 x^{4} - 108)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.15

Add $- 16 x^{9}$ and $12 x^{9}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 48 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 - 54 x^{5} + 30 x^{8} - 111 x^{4} - 108)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.16

Add $48 x^{8}$ and $30 x^{8}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 24 x^{5} + 128 x^{4} - 384 x^{3} + 192 - 54 x^{5} - 111 x^{4} - 108)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.17

Subtract $54 x^{5}$ from $- 24 x^{5}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 128 x^{4} - 384 x^{3} + 192 - 111 x^{4} - 108)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.18

Subtract $111 x^{4}$ from $128 x^{4}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 192 - 108)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.1.34.19

Subtract $108$ from $192$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 84)}{3 x^{\frac{28}{3}}}}{{(- 3 \frac{1}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.2

Combine terms.

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Step 32.2.1

Combine $- 3$ and $\frac{1}{x^{3}}$ .

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 84)}{3 x^{\frac{28}{3}}}}{{(\frac{- 3}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.2.2

Move the negative in front of the fraction.

$\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 84)}{3 x^{\frac{28}{3}}}}{{(- \frac{3}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.2.3

Rewrite $\frac{\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 84)}{3 x^{\frac{28}{3}}}}{{(- \frac{3}{x^{3}} - 2 x + 6)}^{2}}$ as a product.

$\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 84)}{3 x^{\frac{28}{3}}} \cdot \frac{1}{{(- \frac{3}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.2.4

Multiply $\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 84)}{3 x^{\frac{28}{3}}}$ by $\frac{1}{{(- \frac{3}{x^{3}} - 2 x + 6)}^{2}}$ .

$\frac{{(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} \cdot 2^{\frac{4}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 84)}{3 x^{\frac{28}{3}} {(- \frac{3}{x^{3}} - 2 x + 6)}^{2}}$

Step 32.3

Reorder terms.

$\frac{2^{\frac{4}{3}} {(2 x^{5} + 5 x^{4} + 4)}^{\frac{1}{3}} (- 4 x^{9} + 78 x^{8} - 78 x^{5} + 17 x^{4} - 384 x^{3} + 84)}{3 x^{\frac{28}{3}} {(- \frac{3}{x^{3}} - 2 x + 6)}^{2}}$