Calculus Examples

$\frac{\sqrt[3]{4 x^{3} + 8}}{{(x + 2)}^{5}}$

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 $f (x) = \sqrt[3]{4 x^{3} + 8}$ and $g (x) = {(x + 2)}^{5}$ .

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x^{3} + 8}$ as ${(4 x^{3} + 8)}^{\frac{1}{3}}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = 4 x^{3} + 8$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $4 x^{3} + 8$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u_{1}$ with $4 x^{3} + 8$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

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

Step 7.2

Subtract $3$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

Move ${(4 x^{3} + 8)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9

By the Sum Rule, the derivative of $4 x^{3} + 8$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [8]$ .

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

Step 10

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

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

Step 11

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

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

Step 12

Multiply $3$ by $4$ .

$\frac{{(x + 2)}^{5} (\frac{1}{3 {(4 x^{3} + 8)}^{\frac{2}{3}}} (12 x^{2} + \frac{d}{d x} [8])) - \sqrt[3]{4 x^{3} + 8} \frac{d}{d x} [{(x + 2)}^{5}]}{{({(x + 2)}^{5})}^{2}}$

Step 13

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{{(x + 2)}^{5} (\frac{1}{3 {(4 x^{3} + 8)}^{\frac{2}{3}}} (12 x^{2} + 0)) - \sqrt[3]{4 x^{3} + 8} \frac{d}{d x} [{(x + 2)}^{5}]}{{({(x + 2)}^{5})}^{2}}$

Step 14

Simplify terms.

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Step 14.1

Add $12 x^{2}$ and $0$ .

$\frac{{(x + 2)}^{5} (\frac{1}{3 {(4 x^{3} + 8)}^{\frac{2}{3}}} (12 x^{2})) - \sqrt[3]{4 x^{3} + 8} \frac{d}{d x} [{(x + 2)}^{5}]}{{({(x + 2)}^{5})}^{2}}$

Step 14.2

Combine $12$ and $\frac{1}{3 {(4 x^{3} + 8)}^{\frac{2}{3}}}$ .

$\frac{{(x + 2)}^{5} (\frac{12}{3 {(4 x^{3} + 8)}^{\frac{2}{3}}} x^{2}) - \sqrt[3]{4 x^{3} + 8} \frac{d}{d x} [{(x + 2)}^{5}]}{{({(x + 2)}^{5})}^{2}}$

Step 14.3

Combine $\frac{12}{3 {(4 x^{3} + 8)}^{\frac{2}{3}}}$ and $x^{2}$ .

$\frac{{(x + 2)}^{5} \frac{12 x^{2}}{3 {(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 8} \frac{d}{d x} [{(x + 2)}^{5}]}{{({(x + 2)}^{5})}^{2}}$

Step 14.4

Factor $3$ out of $12 x^{2}$ .

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

Step 15

Cancel the common factors.

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Step 15.1

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

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

Step 15.2

Cancel the common factor.

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

Step 15.3

Rewrite the expression.

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

Step 16

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x + 2$ .

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Step 16.1

To apply the Chain Rule, set $u_{2}$ as $x + 2$ .

$\frac{{(x + 2)}^{5} \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 8} (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d x} [x + 2])}{{({(x + 2)}^{5})}^{2}}$

Step 16.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$\frac{{(x + 2)}^{5} \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 8} (5 {u_{2}}^{4} \frac{d}{d x} [x + 2])}{{({(x + 2)}^{5})}^{2}}$

Step 16.3

Replace all occurrences of $u_{2}$ with $x + 2$ .

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

Step 17

Differentiate.

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Step 17.1

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

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

Step 17.2

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

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

Step 17.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 17.4

Simplify the expression.

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Step 17.4.1

Add $1$ and $0$ .

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

Step 17.4.2

Multiply $5$ by $1$ .

$\frac{{(x + 2)}^{5} \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 8} (5 {(x + 2)}^{4})}{{({(x + 2)}^{5})}^{2}}$

Step 18

Simplify.

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Step 18.1

Simplify the numerator.

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Step 18.1.1

Factor ${(x + 2)}^{4}$ out of ${(x + 2)}^{5} \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 8} (5 {(x + 2)}^{4})$ .

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Step 18.1.1.1

Factor ${(x + 2)}^{4}$ out of ${(x + 2)}^{5} \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}$ .

$\frac{{(x + 2)}^{4} ((x + 2) \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}) - \sqrt[3]{4 x^{3} + 8} (5 {(x + 2)}^{4})}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.1.2

Factor ${(x + 2)}^{4}$ out of $- \sqrt[3]{4 x^{3} + 8} (5 {(x + 2)}^{4})$ .

$\frac{{(x + 2)}^{4} ((x + 2) \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}) + {(x + 2)}^{4} (- \sqrt[3]{4 x^{3} + 8} \cdot (5))}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.1.3

Factor ${(x + 2)}^{4}$ out of ${(x + 2)}^{4} ((x + 2) \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}) + {(x + 2)}^{4} (- \sqrt[3]{4 x^{3} + 8} \cdot (5))$ .

$\frac{{(x + 2)}^{4} ((x + 2) \frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 8} \cdot (5))}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.2

Multiply $x + 2$ by $\frac{4 x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}$ .

$\frac{{(x + 2)}^{4} (\frac{(x + 2) (4 x^{2})}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 8} \cdot (5))}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.3

Move $4$ to the left of $x + 2$ .

$\frac{{(x + 2)}^{4} (\frac{4 \cdot (x + 2) x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 8} \cdot (5))}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.4

Factor $4$ out of $4 x^{3} + 8$ .

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Step 18.1.4.1

Factor $4$ out of $4 x^{3}$ .

$\frac{{(x + 2)}^{4} (\frac{4 (x + 2) x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 (x^{3}) + 8} \cdot 5)}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.4.2

Factor $4$ out of $8$ .

$\frac{{(x + 2)}^{4} (\frac{4 (x + 2) x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 x^{3} + 4 \cdot 2} \cdot 5)}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.4.3

Factor $4$ out of $4 x^{3} + 4 \cdot 2$ .

$\frac{{(x + 2)}^{4} (\frac{4 (x + 2) x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - \sqrt[3]{4 (x^{3} + 2)} \cdot 5)}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.5

Multiply $5$ by $- 1$ .

$\frac{{(x + 2)}^{4} (\frac{4 (x + 2) x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - 5 \sqrt[3]{4 (x^{3} + 2)})}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.6

To write $- 5 \sqrt[3]{4 (x^{3} + 2)}$ as a fraction with a common denominator, multiply by $\frac{{(4 x^{3} + 8)}^{\frac{2}{3}}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}$ .

$\frac{{(x + 2)}^{4} (\frac{4 (x + 2) x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} - 5 \sqrt[3]{4 (x^{3} + 2)} \cdot \frac{{(4 x^{3} + 8)}^{\frac{2}{3}}}{{(4 x^{3} + 8)}^{\frac{2}{3}}})}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.7

Combine $- 5 \sqrt[3]{4 (x^{3} + 2)}$ and $\frac{{(4 x^{3} + 8)}^{\frac{2}{3}}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}$ .

$\frac{{(x + 2)}^{4} (\frac{4 (x + 2) x^{2}}{{(4 x^{3} + 8)}^{\frac{2}{3}}} + \frac{- 5 \sqrt[3]{4 (x^{3} + 2)} {(4 x^{3} + 8)}^{\frac{2}{3}}}{{(4 x^{3} + 8)}^{\frac{2}{3}}})}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.8

Combine the numerators over the common denominator.

$\frac{{(x + 2)}^{4} \frac{4 (x + 2) x^{2} - 5 \sqrt[3]{4 (x^{3} + 2)} {(4 x^{3} + 8)}^{\frac{2}{3}}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.9

Rewrite $\frac{4 (x + 2) x^{2} - 5 \sqrt[3]{4 (x^{3} + 2)} {(4 x^{3} + 8)}^{\frac{2}{3}}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}$ in a factored form.

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Step 18.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 (x^{3} + 2)}$ as ${(4 (x^{3} + 2))}^{\frac{1}{3}}$ .

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

Step 18.1.9.2

Apply the distributive property.

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

Step 18.1.9.3

Multiply $4$ by $2$ .

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

Step 18.1.9.4

Apply the distributive property.

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

Step 18.1.9.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 18.1.9.5.1

Move $x^{2}$ .

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

Step 18.1.9.5.2

Multiply $x^{2}$ by $x$ .

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Step 18.1.9.5.2.1

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

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

Step 18.1.9.5.2.2

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

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

Step 18.1.9.5.3

Add $2$ and $1$ .

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

Step 18.1.9.6

Apply the distributive property.

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

Step 18.1.9.7

Multiply $4$ by $2$ .

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

Step 18.1.9.8

Multiply ${(4 x^{3} + 8)}^{\frac{1}{3}}$ by ${(4 x^{3} + 8)}^{\frac{2}{3}}$ by adding the exponents.

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Step 18.1.9.8.1

Move ${(4 x^{3} + 8)}^{\frac{2}{3}}$ .

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

Step 18.1.9.8.2

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

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

Step 18.1.9.8.3

Combine the numerators over the common denominator.

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

Step 18.1.9.8.4

Add $2$ and $1$ .

$\frac{{(x + 2)}^{4} \frac{4 x^{3} + 8 x^{2} - 5 {(4 x^{3} + 8)}^{\frac{3}{3}}}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.9.8.5

Divide $3$ by $3$ .

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

Step 18.1.9.9

Simplify $- 5 {(4 x^{3} + 8)}^{1}$ .

$\frac{{(x + 2)}^{4} \frac{4 x^{3} + 8 x^{2} - 5 (4 x^{3} + 8)}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.9.10

Apply the distributive property.

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

Step 18.1.9.11

Multiply $4$ by $- 5$ .

$\frac{{(x + 2)}^{4} \frac{4 x^{3} + 8 x^{2} - 20 x^{3} - 5 \cdot 8}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.9.12

Multiply $- 5$ by $8$ .

$\frac{{(x + 2)}^{4} \frac{4 x^{3} + 8 x^{2} - 20 x^{3} - 40}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.9.13

Subtract $20 x^{3}$ from $4 x^{3}$ .

$\frac{{(x + 2)}^{4} \frac{- 16 x^{3} + 8 x^{2} - 40}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.9.14

Factor $8$ out of $- 16 x^{3} + 8 x^{2} - 40$ .

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Step 18.1.9.14.1

Factor $8$ out of $- 16 x^{3}$ .

$\frac{{(x + 2)}^{4} \frac{8 (- 2 x^{3}) + 8 x^{2} - 40}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.9.14.2

Factor $8$ out of $8 x^{2}$ .

$\frac{{(x + 2)}^{4} \frac{8 (- 2 x^{3}) + 8 (x^{2}) - 40}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.1.9.14.3

Factor $8$ out of $- 40$ .

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

Step 18.1.9.14.4

Factor $8$ out of $8 (- 2 x^{3}) + 8 x^{2}$ .

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

Step 18.1.9.14.5

Factor $8$ out of $8 (- 2 x^{3} + x^{2}) + 8 \cdot - 5$ .

$\frac{{(x + 2)}^{4} \frac{8 (- 2 x^{3} + x^{2} - 5)}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.2

Combine terms.

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Step 18.2.1

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

$\frac{\frac{{(x + 2)}^{4} (8 (- 2 x^{3} + x^{2} - 5))}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{({(x + 2)}^{5})}^{2}}$

Step 18.2.2

Move $8$ to the left of ${(x + 2)}^{4}$ .

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

Step 18.2.3

Multiply the exponents in ${({(x + 2)}^{5})}^{2}$ .

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Step 18.2.3.1

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

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

Step 18.2.3.2

Multiply $5$ by $2$ .

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

Step 18.2.4

Rewrite $\frac{\frac{8 {(x + 2)}^{4} (- 2 x^{3} + x^{2} - 5)}{{(4 x^{3} + 8)}^{\frac{2}{3}}}}{{(x + 2)}^{10}}$ as a product.

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

Step 18.2.5

Multiply $\frac{8 {(x + 2)}^{4} (- 2 x^{3} + x^{2} - 5)}{{(4 x^{3} + 8)}^{\frac{2}{3}}}$ by $\frac{1}{{(x + 2)}^{10}}$ .

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

Step 18.2.6

Factor ${(x + 2)}^{4}$ out of $8 {(x + 2)}^{4} (- 2 x^{3} + x^{2} - 5)$ .

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

Step 18.2.7

Cancel the common factors.

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Step 18.2.7.1

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

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

Step 18.2.7.2

Cancel the common factor.

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

Step 18.2.7.3

Rewrite the expression.

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

Step 18.3

Factor $- 1$ out of $- 2 x^{3}$ .

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

Step 18.4

Factor $- 1$ out of $x^{2}$ .

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

Step 18.5

Factor $- 1$ out of $- (2 x^{3}) - 1 (- x^{2})$ .

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

Step 18.6

Rewrite $- 5$ as $- 1 (5)$ .

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

Step 18.7

Factor $- 1$ out of $- (2 x^{3} - x^{2}) - 1 (5)$ .

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

Step 18.8

Rewrite $- (2 x^{3} - x^{2} + 5)$ as $- 1 (2 x^{3} - x^{2} + 5)$ .

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

Step 18.9

Move the negative in front of the fraction.

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