Algebra Examples

$y = 2 \sqrt[5]{x}$

Step 1

Interchange the variables.

$x = 2 \sqrt[5]{y}$

Step 2

Solve for $y$ .

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Step 2.1

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

$2 \sqrt[5]{y} = x$

Step 2.2

Divide each term in $2 \sqrt[5]{y} = x$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 \sqrt[5]{y} = x$ by $2$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $\sqrt[5]{y}$ by $1$ .

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

Step 2.3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

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

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

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

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Step 2.4.2.1.1.1

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

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

Step 2.4.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

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

Step 2.4.2.1.1.2.2

Rewrite the expression.

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

Step 2.4.2.1.2

Simplify.

$y = {(\frac{x}{2})}^{5}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(\frac{x}{2})}^{5}$ .

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Step 2.4.3.1.1

Apply the product rule to $\frac{x}{2}$ .

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

Step 2.4.3.1.2

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

$y = \frac{x^{5}}{32}$

Step 3

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

$f^{- 1} (x) = \frac{x^{5}}{32}$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{5}}{32}$ is the inverse of $f (x) = 2 \sqrt[5]{x}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Apply the product rule to $2 \sqrt[5]{x}$ .

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

Step 4.2.3.2

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

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

Step 4.2.3.3

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

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Step 4.2.3.3.1

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

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

Step 4.2.3.3.2

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

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

Step 4.2.3.3.3

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

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

Step 4.2.3.3.4

Cancel the common factor of $5$ .

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Step 4.2.3.3.4.1

Cancel the common factor.

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

Step 4.2.3.3.4.2

Rewrite the expression.

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

Step 4.2.3.3.5

Simplify.

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

Step 4.2.4

Cancel the common factor of $32$ .

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

Divide $x$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x^{5}}{32})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Remove parentheses.

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

Step 4.3.4

Rewrite $32$ as $2^{5}$ .

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

Step 4.3.5

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

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

Step 4.3.6

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

$f (\frac{x^{5}}{32}) = 2 (\frac{x}{2})$

Step 4.3.7

Cancel the common factor of $2$ .

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Step 4.3.7.1

Cancel the common factor.

$f (\frac{x^{5}}{32}) = 2 (\frac{x}{2})$

Step 4.3.7.2

Rewrite the expression.

$f (\frac{x^{5}}{32}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{5}}{32}$ is the inverse of $f (x) = 2 \sqrt[5]{x}$ .

$f^{- 1} (x) = \frac{x^{5}}{32}$