Calculus Examples

$y = x^{\frac{2}{3}} (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7$ .

$x^{\frac{2}{3}} \frac{d}{d x} [6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7] + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7$ with respect to $x$ is $\frac{d}{d x} [6 x^{\frac{4}{3}}] + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]$ .

$x^{\frac{2}{3}} (\frac{d}{d x} [6 x^{\frac{4}{3}}] + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{\frac{4}{3}}$ with respect to $x$ is $6 \frac{d}{d x} [x^{\frac{4}{3}}]$ .

$x^{\frac{2}{3}} (6 \frac{d}{d x} [x^{\frac{4}{3}}] + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.3

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

$x^{\frac{2}{3}} (6 (\frac{4}{3} x^{\frac{4}{3} - 1}) + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 3

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

$x^{\frac{2}{3}} (6 (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 4

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

$x^{\frac{2}{3}} (6 (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 5

Combine the numerators over the common denominator.

$x^{\frac{2}{3}} (6 (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}}) + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$x^{\frac{2}{3}} (6 (\frac{4}{3} x^{\frac{4 - 3}{3}}) + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 6.2

Subtract $3$ from $4$ .

$x^{\frac{2}{3}} (6 (\frac{4}{3} x^{\frac{1}{3}}) + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 7

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

$x^{\frac{2}{3}} (6 \frac{4 x^{\frac{1}{3}}}{3} + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8

Combine $6$ and $\frac{4 x^{\frac{1}{3}}}{3}$ .

$x^{\frac{2}{3}} (\frac{6 (4 x^{\frac{1}{3}})}{3} + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 9

Multiply $4$ by $6$ .

$x^{\frac{2}{3}} (\frac{24 x^{\frac{1}{3}}}{3} + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 10

Factor $3$ out of $24 x^{\frac{1}{3}}$ .

$x^{\frac{2}{3}} (\frac{3 (8 x^{\frac{1}{3}})}{3} + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 11

Cancel the common factors.

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Step 11.1

Factor $3$ out of $3$ .

$x^{\frac{2}{3}} (\frac{3 (8 x^{\frac{1}{3}})}{3 (1)} + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 11.2

Cancel the common factor.

$x^{\frac{2}{3}} (\frac{3 (8 x^{\frac{1}{3}})}{3 \cdot 1} + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 11.3

Rewrite the expression.

$x^{\frac{2}{3}} (\frac{8 x^{\frac{1}{3}}}{1} + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 11.4

Divide $8 x^{\frac{1}{3}}$ by $1$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{d}{d x} [2 x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 12

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{7}{3}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{7}{3}}]$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + 2 \frac{d}{d x} [x^{\frac{7}{3}}] + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{3}$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + 2 (\frac{7}{3} x^{\frac{7}{3} - 1}) + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 14

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

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + 2 (\frac{7}{3} x^{\frac{7}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 15

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

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + 2 (\frac{7}{3} x^{\frac{7}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 16

Combine the numerators over the common denominator.

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + 2 (\frac{7}{3} x^{\frac{7 - 1 \cdot 3}{3}}) + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 17

Simplify the numerator.

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Step 17.1

Multiply $- 1$ by $3$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + 2 (\frac{7}{3} x^{\frac{7 - 3}{3}}) + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 17.2

Subtract $3$ from $7$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + 2 (\frac{7}{3} x^{\frac{4}{3}}) + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 18

Combine fractions.

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Step 18.1

Combine $\frac{7}{3}$ and $x^{\frac{4}{3}}$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + 2 \frac{7 x^{\frac{4}{3}}}{3} + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 18.2

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

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{2 (7 x^{\frac{4}{3}})}{3} + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 18.3

Multiply $7$ by $2$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3} + \frac{d}{d x} [7]) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 19

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3} + 0) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 20

Add $8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}$ and $0$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 21

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 22

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

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 23

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

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 24

Combine the numerators over the common denominator.

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 25

Simplify the numerator.

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Step 25.1

Multiply $- 1$ by $3$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 25.2

Subtract $3$ from $2$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 26

Move the negative in front of the fraction.

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) (\frac{2}{3} x^{- \frac{1}{3}})$

Step 27

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{2 x^{- \frac{1}{3}}}{3}$

Step 28

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}} + \frac{14 x^{\frac{4}{3}}}{3}) + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{2}{3 x^{\frac{1}{3}}}$

Step 29

Simplify.

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Step 29.1

Apply the distributive property.

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}}) + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + (6 x^{\frac{4}{3}} + 2 x^{\frac{7}{3}} + 7) \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.2

Apply the distributive property.

$x^{\frac{2}{3}} (8 x^{\frac{1}{3}}) + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3

Combine terms.

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Step 29.3.1

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

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Step 29.3.1.1

Move $x^{\frac{1}{3}}$ .

$x^{\frac{1}{3}} x^{\frac{2}{3}} \cdot 8 + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.1.2

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

$x^{\frac{1}{3} + \frac{2}{3}} \cdot 8 + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.1.3

Combine the numerators over the common denominator.

$x^{\frac{1 + 2}{3}} \cdot 8 + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.1.4

Add $1$ and $2$ .

$x^{\frac{3}{3}} \cdot 8 + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.1.5

Divide $3$ by $3$ .

$x^{1} \cdot 8 + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.2

Simplify $x^{1} \cdot 8$ .

$x \cdot 8 + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.3

Move $8$ to the left of $x$ .

$8 \cdot x + x^{\frac{2}{3}} \frac{14 x^{\frac{4}{3}}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.4

Combine $x^{\frac{2}{3}}$ and $\frac{14 x^{\frac{4}{3}}}{3}$ .

$8 x + \frac{x^{\frac{2}{3}} (14 x^{\frac{4}{3}})}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.5

Multiply $x^{\frac{2}{3}}$ by $x^{\frac{4}{3}}$ by adding the exponents.

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Step 29.3.5.1

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

$8 x + \frac{x^{\frac{4}{3}} x^{\frac{2}{3}} \cdot 14}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.5.2

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

$8 x + \frac{x^{\frac{4}{3} + \frac{2}{3}} \cdot 14}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.5.3

Combine the numerators over the common denominator.

$8 x + \frac{x^{\frac{4 + 2}{3}} \cdot 14}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.5.4

Add $4$ and $2$ .

$8 x + \frac{x^{\frac{6}{3}} \cdot 14}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.5.5

Divide $6$ by $3$ .

$8 x + \frac{x^{2} \cdot 14}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.6

Move $14$ to the left of $x^{2}$ .

$8 x + \frac{14 \cdot x^{2}}{3} + 6 x^{\frac{4}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.7

Combine $6$ and $\frac{2}{3 x^{\frac{1}{3}}}$ .

$8 x + \frac{14 x^{2}}{3} + x^{\frac{4}{3}} \frac{6 \cdot 2}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.8

Multiply $6$ by $2$ .

$8 x + \frac{14 x^{2}}{3} + x^{\frac{4}{3}} \frac{12}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.9

Combine $x^{\frac{4}{3}}$ and $\frac{12}{3 x^{\frac{1}{3}}}$ .

$8 x + \frac{14 x^{2}}{3} + \frac{x^{\frac{4}{3}} \cdot 12}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.10

Move $12$ to the left of $x^{\frac{4}{3}}$ .

$8 x + \frac{14 x^{2}}{3} + \frac{12 \cdot x^{\frac{4}{3}}}{3 x^{\frac{1}{3}}} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.11

Move $x^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$8 x + \frac{14 x^{2}}{3} + \frac{12 x^{\frac{4}{3}} x^{- \frac{1}{3}}}{3} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.12

Simplify the numerator.

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Step 29.3.12.1

Multiply $x^{\frac{4}{3}}$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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Step 29.3.12.1.1

Move $x^{- \frac{1}{3}}$ .

$8 x + \frac{14 x^{2}}{3} + \frac{12 (x^{- \frac{1}{3}} x^{\frac{4}{3}})}{3} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.12.1.2

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

$8 x + \frac{14 x^{2}}{3} + \frac{12 x^{- \frac{1}{3} + \frac{4}{3}}}{3} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.12.1.3

Combine the numerators over the common denominator.

$8 x + \frac{14 x^{2}}{3} + \frac{12 x^{\frac{- 1 + 4}{3}}}{3} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.12.1.4

Add $- 1$ and $4$ .

$8 x + \frac{14 x^{2}}{3} + \frac{12 x^{\frac{3}{3}}}{3} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.12.1.5

Divide $3$ by $3$ .

$8 x + \frac{14 x^{2}}{3} + \frac{12 x^{1}}{3} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.12.2

Simplify $12 x^{1}$ .

$8 x + \frac{14 x^{2}}{3} + \frac{12 x}{3} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.13

Cancel the common factor of $12$ and $3$ .

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Step 29.3.13.1

Factor $3$ out of $12 x$ .

$8 x + \frac{14 x^{2}}{3} + \frac{3 (4 x)}{3} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.13.2

Cancel the common factors.

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Step 29.3.13.2.1

Factor $3$ out of $3$ .

$8 x + \frac{14 x^{2}}{3} + \frac{3 (4 x)}{3 (1)} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.13.2.2

Cancel the common factor.

$8 x + \frac{14 x^{2}}{3} + \frac{3 (4 x)}{3 \cdot 1} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.13.2.3

Rewrite the expression.

$8 x + \frac{14 x^{2}}{3} + \frac{4 x}{1} + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.13.2.4

Divide $4 x$ by $1$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + 2 x^{\frac{7}{3}} \frac{2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.14

Combine $2$ and $\frac{2}{3 x^{\frac{1}{3}}}$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + x^{\frac{7}{3}} \frac{2 \cdot 2}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.15

Multiply $2$ by $2$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + x^{\frac{7}{3}} \frac{4}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.16

Combine $x^{\frac{7}{3}}$ and $\frac{4}{3 x^{\frac{1}{3}}}$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{x^{\frac{7}{3}} \cdot 4}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.17

Move $4$ to the left of $x^{\frac{7}{3}}$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 \cdot x^{\frac{7}{3}}}{3 x^{\frac{1}{3}}} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.18

Move $x^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 x^{\frac{7}{3}} x^{- \frac{1}{3}}}{3} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.19

Multiply $x^{\frac{7}{3}}$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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Step 29.3.19.1

Move $x^{- \frac{1}{3}}$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 (x^{- \frac{1}{3}} x^{\frac{7}{3}})}{3} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.19.2

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

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 x^{- \frac{1}{3} + \frac{7}{3}}}{3} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.19.3

Combine the numerators over the common denominator.

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 x^{\frac{- 1 + 7}{3}}}{3} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.19.4

Add $- 1$ and $7$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 x^{\frac{6}{3}}}{3} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.19.5

Divide $6$ by $3$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 x^{2}}{3} + 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 29.3.20

Combine $7$ and $\frac{2}{3 x^{\frac{1}{3}}}$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 x^{2}}{3} + \frac{7 \cdot 2}{3 x^{\frac{1}{3}}}$

Step 29.3.21

Multiply $7$ by $2$ .

$8 x + \frac{14 x^{2}}{3} + 4 x + \frac{4 x^{2}}{3} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.3.22

Add $8 x$ and $4 x$ .

$\frac{14 x^{2}}{3} + 12 x + \frac{4 x^{2}}{3} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.3.23

Combine the numerators over the common denominator.

$12 x + \frac{14 x^{2} + 4 x^{2}}{3} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.3.24

Add $14 x^{2}$ and $4 x^{2}$ .

$12 x + \frac{18 x^{2}}{3} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.3.25

Cancel the common factor of $18$ and $3$ .

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Step 29.3.25.1

Factor $3$ out of $18 x^{2}$ .

$12 x + \frac{3 (6 x^{2})}{3} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.3.25.2

Cancel the common factors.

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Step 29.3.25.2.1

Factor $3$ out of $3$ .

$12 x + \frac{3 (6 x^{2})}{3 (1)} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.3.25.2.2

Cancel the common factor.

$12 x + \frac{3 (6 x^{2})}{3 \cdot 1} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.3.25.2.3

Rewrite the expression.

$12 x + \frac{6 x^{2}}{1} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.3.25.2.4

Divide $6 x^{2}$ by $1$ .

$12 x + 6 x^{2} + \frac{14}{3 x^{\frac{1}{3}}}$

Step 29.4

Reorder terms.

$6 x^{2} + 12 x + \frac{14}{3 x^{\frac{1}{3}}}$