Algebra Examples

$f (x) = \sqrt[3]{2 x^{5} - 10}$

Step 1

Write $f (x) = \sqrt[3]{2 x^{5} - 10}$ as an equation.

$y = \sqrt[3]{2 x^{5} - 10}$

Step 2

Interchange the variables.

$x = \sqrt[3]{2 y^{5} - 10}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[3]{2 y^{5} - 10} = x$ .

$\sqrt[3]{2 y^{5} - 10} = x$

Step 3.2

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{2 y^{5} - 10}}^{3} = x^{3}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 y^{5} - 10}$ as ${(2 y^{5} - 10)}^{\frac{1}{3}}$ .

${({(2 y^{5} - 10)}^{\frac{1}{3}})}^{3} = x^{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(2 y^{5} - 10)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(2 y^{5} - 10)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.1.1.1

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

${(2 y^{5} - 10)}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 3.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(2 y^{5} - 10)}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(2 y^{5} - 10)}^{1} = x^{3}$

Step 3.3.2.1.2

Simplify.

$2 y^{5} - 10 = x^{3}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $10$ to both sides of the equation.

$2 y^{5} = x^{3} + 10$

Step 3.4.2

Divide each term in $2 y^{5} = x^{3} + 10$ by $2$ and simplify.

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Step 3.4.2.1

Divide each term in $2 y^{5} = x^{3} + 10$ by $2$ .

$\frac{2 y^{5}}{2} = \frac{x^{3}}{2} + \frac{10}{2}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{2 y^{5}}{2} = \frac{x^{3}}{2} + \frac{10}{2}$

Step 3.4.2.2.1.2

Divide $y^{5}$ by $1$ .

$y^{5} = \frac{x^{3}}{2} + \frac{10}{2}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $10$ by $2$ .

$y^{5} = \frac{x^{3}}{2} + 5$

Step 3.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[5]{\frac{x^{3}}{2} + 5}$

Step 3.4.4

Simplify $\sqrt[5]{\frac{x^{3}}{2} + 5}$ .

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Step 3.4.4.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt[5]{\frac{x^{3}}{2} + 5 \cdot \frac{2}{2}}$

Step 3.4.4.2

Combine $5$ and $\frac{2}{2}$ .

$y = \sqrt[5]{\frac{x^{3}}{2} + \frac{5 \cdot 2}{2}}$

Step 3.4.4.3

Combine the numerators over the common denominator.

$y = \sqrt[5]{\frac{x^{3} + 5 \cdot 2}{2}}$

Step 3.4.4.4

Multiply $5$ by $2$ .

$y = \sqrt[5]{\frac{x^{3} + 10}{2}}$

Step 3.4.4.5

Rewrite $\sqrt[5]{\frac{x^{3} + 10}{2}}$ as $\frac{\sqrt[5]{x^{3} + 10}}{\sqrt[5]{2}}$ .

$y = \frac{\sqrt[5]{x^{3} + 10}}{\sqrt[5]{2}}$

Step 3.4.4.6

Multiply $\frac{\sqrt[5]{x^{3} + 10}}{\sqrt[5]{2}}$ by $\frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$ .

$y = \frac{\sqrt[5]{x^{3} + 10}}{\sqrt[5]{2}} \cdot \frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$

Step 3.4.4.7

Combine and simplify the denominator.

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Step 3.4.4.7.1

Multiply $\frac{\sqrt[5]{x^{3} + 10}}{\sqrt[5]{2}}$ by $\frac{{\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{4}}$ .

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{\sqrt[5]{2} {\sqrt[5]{2}}^{4}}$

Step 3.4.4.7.2

Raise $\sqrt[5]{2}$ to the power of $1$ .

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{1} {\sqrt[5]{2}}^{4}}$

Step 3.4.4.7.3

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

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{1 + 4}}$

Step 3.4.4.7.4

Add $1$ and $4$ .

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{{\sqrt[5]{2}}^{5}}$

Step 3.4.4.7.5

Rewrite ${\sqrt[5]{2}}^{5}$ as $2$ .

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Step 3.4.4.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2}$ as $2^{\frac{1}{5}}$ .

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{{(2^{\frac{1}{5}})}^{5}}$

Step 3.4.4.7.5.2

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

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{2^{\frac{1}{5} \cdot 5}}$

Step 3.4.4.7.5.3

Combine $\frac{1}{5}$ and $5$ .

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{2^{\frac{5}{5}}}$

Step 3.4.4.7.5.4

Cancel the common factor of $5$ .

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Step 3.4.4.7.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{2^{\frac{5}{5}}}$

Step 3.4.4.7.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{2^{1}}$

Step 3.4.4.7.5.5

Evaluate the exponent.

$y = \frac{\sqrt[5]{x^{3} + 10} {\sqrt[5]{2}}^{4}}{2}$

Step 3.4.4.8

Simplify the numerator.

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Step 3.4.4.8.1

Rewrite ${\sqrt[5]{2}}^{4}$ as $\sqrt[5]{2^{4}}$ .

$y = \frac{\sqrt[5]{x^{3} + 10} \sqrt[5]{2^{4}}}{2}$

Step 3.4.4.8.2

Raise $2$ to the power of $4$ .

$y = \frac{\sqrt[5]{x^{3} + 10} \sqrt[5]{16}}{2}$

Step 3.4.4.9

Simplify with factoring out.

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Step 3.4.4.9.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[5]{(x^{3} + 10) \cdot 16}}{2}$

Step 3.4.4.9.2

Reorder factors in $\frac{\sqrt[5]{(x^{3} + 10) \cdot 16}}{2}$ .

$y = \frac{\sqrt[5]{16 (x^{3} + 10)}}{2}$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{\sqrt[5]{16 (x^{3} + 10)}}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[5]{16 (x^{3} + 10)}}{2}$ is the inverse of $f (x) = \sqrt[3]{2 x^{5} - 10}$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (\sqrt[3]{2 x^{5} - 10})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{16 ({(\sqrt[3]{2 x^{5} - 10})}^{3} + 10)}}{2}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x^{5} - 10}$ as ${(2 x^{5} - 10)}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{16 ({({(2 x^{5} - 10)}^{\frac{1}{3}})}^{3} + 10)}}{2}$

Step 5.2.3.2

Multiply the exponents in ${({(2 x^{5} - 10)}^{\frac{1}{3}})}^{3}$ .

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Step 5.2.3.2.1

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

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{16 ({(2 x^{5} - 10)}^{\frac{1}{3} \cdot 3} + 10)}}{2}$

Step 5.2.3.2.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.2.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{16 ({(2 x^{5} - 10)}^{\frac{1}{3} \cdot 3} + 10)}}{2}$

Step 5.2.3.2.2.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{16 ((2 x^{5} - 10) + 10)}}{2}$

Step 5.2.3.3

Simplify.

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{16 (2 x^{5} - 10 + 10)}}{2}$

Step 5.2.3.4

Add $- 10$ and $10$ .

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{16 (2 x^{5} + 0)}}{2}$

Step 5.2.3.5

Add $2 x^{5}$ and $0$ .

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{16 \cdot (2 x^{5})}}{2}$

Step 5.2.3.6

Multiply $16$ by $2$ .

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{32 x^{5}}}{2}$

Step 5.2.3.7

Rewrite $32 x^{5}$ as ${(2 x)}^{5}$ .

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{\sqrt[5]{{(2 x)}^{5}}}{2}$

Step 5.2.3.8

Pull terms out from under the radical, assuming real numbers.

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{2 x}{2}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = \frac{2 x}{2}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (\sqrt[3]{2 x^{5} - 10}) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 {(\frac{\sqrt[5]{16 (x^{3} + 10)}}{2})}^{5} - 10}$

Step 5.3.3

Simplify with factoring out.

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Step 5.3.3.1

Factor $2$ out of $2 {(\frac{\sqrt[5]{16 (x^{3} + 10)}}{2})}^{5} - 10$ .

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Step 5.3.3.1.1

Factor $2$ out of $2 {(\frac{\sqrt[5]{16 (x^{3} + 10)}}{2})}^{5}$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 ({(\frac{\sqrt[5]{16 (x^{3} + 10)}}{2})}^{5}) - 10}$

Step 5.3.3.1.2

Factor $2$ out of $- 10$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 {(\frac{\sqrt[5]{16 (x^{3} + 10)}}{2})}^{5} + 2 \cdot - 5}$

Step 5.3.3.1.3

Factor $2$ out of $2 {(\frac{\sqrt[5]{16 (x^{3} + 10)}}{2})}^{5} + 2 \cdot - 5$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 ({(\frac{\sqrt[5]{16 (x^{3} + 10)}}{2})}^{5} - 5)}$

Step 5.3.3.2

Apply the product rule to $\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{{\sqrt[5]{16 (x^{3} + 10)}}^{5}}{2^{5}} - 5)}$

Step 5.3.4

Simplify the numerator.

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Step 5.3.4.1

Rewrite ${\sqrt[5]{16 (x^{3} + 10)}}^{5}$ as $16 (x^{3} + 10)$ .

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Step 5.3.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{16 (x^{3} + 10)}$ as ${(16 (x^{3} + 10))}^{\frac{1}{5}}$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{{({(16 (x^{3} + 10))}^{\frac{1}{5}})}^{5}}{2^{5}} - 5)}$

Step 5.3.4.1.2

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

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{{(16 (x^{3} + 10))}^{\frac{1}{5} \cdot 5}}{2^{5}} - 5)}$

Step 5.3.4.1.3

Combine $\frac{1}{5}$ and $5$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{{(16 (x^{3} + 10))}^{\frac{5}{5}}}{2^{5}} - 5)}$

Step 5.3.4.1.4

Cancel the common factor of $5$ .

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Step 5.3.4.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{{(16 (x^{3} + 10))}^{\frac{5}{5}}}{2^{5}} - 5)}$

Step 5.3.4.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 (x^{3} + 10)}{2^{5}} - 5)}$

Step 5.3.4.1.5

Simplify.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 (x^{3} + 10)}{2^{5}} - 5)}$

Step 5.3.4.2

Apply the distributive property.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 x^{3} + 16 \cdot 10}{2^{5}} - 5)}$

Step 5.3.4.3

Multiply $16$ by $10$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 x^{3} + 160}{2^{5}} - 5)}$

Step 5.3.4.4

Factor $16$ out of $16 x^{3} + 160$ .

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Step 5.3.4.4.1

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

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 (x^{3}) + 160}{2^{5}} - 5)}$

Step 5.3.4.4.2

Factor $16$ out of $160$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 x^{3} + 16 \cdot 10}{2^{5}} - 5)}$

Step 5.3.4.4.3

Factor $16$ out of $16 x^{3} + 16 \cdot 10$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 (x^{3} + 10)}{2^{5}} - 5)}$

Step 5.3.5

Raise $2$ to the power of $5$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 (x^{3} + 10)}{32} - 5)}$

Step 5.3.6

Cancel the common factors.

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Step 5.3.6.1

Factor $16$ out of $32$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 (x^{3} + 10)}{16 \cdot 2} - 5)}$

Step 5.3.6.2

Cancel the common factor.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{16 (x^{3} + 10)}{16 \cdot 2} - 5)}$

Step 5.3.6.3

Rewrite the expression.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{x^{3} + 10}{2} - 5)}$

Step 5.3.7

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{x^{3} + 10}{2} - 5 \cdot \frac{2}{2})}$

Step 5.3.8

Combine $- 5$ and $\frac{2}{2}$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{x^{3} + 10}{2} + \frac{- 5 \cdot 2}{2})}$

Step 5.3.9

Combine the numerators over the common denominator.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{x^{3} + 10 - 5 \cdot 2}{2})}$

Step 5.3.10

Rewrite $\frac{x^{3} + 10 - 5 \cdot 2}{2}$ in a factored form.

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Step 5.3.10.1

Multiply $- 5$ by $2$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{x^{3} + 10 - 10}{2})}$

Step 5.3.10.2

Subtract $10$ from $10$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{x^{3} + 0}{2})}$

Step 5.3.10.3

Add $x^{3}$ and $0$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{2 (\frac{x^{3}}{2})}$

Step 5.3.11

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

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{\frac{2 x^{3}}{2}}$

Step 5.3.12

Reduce the expression by cancelling the common factors.

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Step 5.3.12.1

Reduce the expression $\frac{2 x^{3}}{2}$ by cancelling the common factors.

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Step 5.3.12.1.1

Cancel the common factor.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{\frac{2 x^{3}}{2}}$

Step 5.3.12.1.2

Rewrite the expression.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{\frac{x^{3}}{1}}$

Step 5.3.12.2

Divide $x^{3}$ by $1$ .

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = \sqrt[3]{x^{3}}$

Step 5.3.13

Pull terms out from under the radical, assuming real numbers.

$f (\frac{\sqrt[5]{16 (x^{3} + 10)}}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[5]{16 (x^{3} + 10)}}{2}$ is the inverse of $f (x) = \sqrt[3]{2 x^{5} - 10}$ .

$f^{- 1} (x) = \frac{\sqrt[5]{16 (x^{3} + 10)}}{2}$