Algebra Examples

$f (x) = {(x^{3} + 5)}^{\frac{1}{5}}$

Step 1

Write $f (x) = {(x^{3} + 5)}^{\frac{1}{5}}$ as an equation.

$y = {(x^{3} + 5)}^{\frac{1}{5}}$

Step 2

Interchange the variables.

$x = {(y^{3} + 5)}^{\frac{1}{5}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(y^{3} + 5)}^{\frac{1}{5}} = x$ .

${(y^{3} + 5)}^{\frac{1}{5}} = x$

Step 3.2

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${({(y^{3} + 5)}^{\frac{1}{5}})}^{5} = x^{5}$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify ${({(y^{3} + 5)}^{\frac{1}{5}})}^{5}$ .

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

${(y^{3} + 5)}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 3.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

${(y^{3} + 5)}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 3.3.1.1.2.2

Rewrite the expression.

${(y^{3} + 5)}^{1} = x^{5}$

Step 3.3.1.2

Simplify.

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

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $5$ from both sides of the equation.

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

Step 3.4.2

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

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

Step 4

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

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

Step 5

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

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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 $f^{- 1} ({(x^{3} + 5)}^{\frac{1}{5}})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

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

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Step 5.2.3.1

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

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

Step 5.2.3.2

Cancel the common factor of $5$ .

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Step 5.2.3.2.1

Cancel the common factor.

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

Step 5.2.3.2.2

Rewrite the expression.

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

Step 5.2.4

Simplify.

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

Step 5.2.5

Subtract $5$ from $5$ .

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

Step 5.2.6

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

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

Step 5.2.7

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

$f^{- 1} ({(x^{3} + 5)}^{\frac{1}{5}}) = x$

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

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

Step 5.3.3

Rewrite ${\sqrt[3]{x^{5} - 5}}^{3}$ as $x^{5} - 5$ .

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Step 5.3.3.1

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

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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

Step 5.3.3.4

Cancel the common factor of $3$ .

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Step 5.3.3.4.1

Cancel the common factor.

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

Step 5.3.3.4.2

Rewrite the expression.

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

Step 5.3.3.5

Simplify.

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

Step 5.3.4

Simplify by adding terms.

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Step 5.3.4.1

Combine the opposite terms in $x^{5} - 5 + 5$ .

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Step 5.3.4.1.1

Add $- 5$ and $5$ .

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

Step 5.3.4.1.2

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

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

Step 5.3.4.2

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

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Step 5.3.4.2.1

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

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

Step 5.3.4.2.2

Cancel the common factor of $5$ .

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Step 5.3.4.2.2.1

Cancel the common factor.

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

Step 5.3.4.2.2.2

Rewrite the expression.

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

Step 5.4

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

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