Algebra Examples

f (x) = 4 \sqrt[5]{\frac{x^{7}}{7}}

$f (x) = 4 \sqrt[5]{\frac{x^{7}}{7}}$

Step 1

Write

$f (x) = 4 \sqrt[5]{\frac{x^{7}}{7}}$ as an equation.

$y = 4 \sqrt[5]{\frac{x^{7}}{7}}$

Step 2

Interchange the variables.

$x = 4 \sqrt[5]{\frac{y^{7}}{7}}$

Step 3

Solve for

$y$ .

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Step 3.1

Rewrite the equation as

$4 \sqrt[5]{\frac{y^{7}}{7}} = x$ .

$4 \sqrt[5]{\frac{y^{7}}{7}} = x$

Step 3.2

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

$5$ .

${(4 \sqrt[5]{\frac{y^{7}}{7}})}^{5} = x^{5}$

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

$\sqrt[5]{\frac{y^{7}}{7}}$ as

${(\frac{y^{7}}{7})}^{\frac{1}{5}}$ .

${(4 {(\frac{y^{7}}{7})}^{\frac{1}{5}})}^{5} = x^{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify

${(4 {(\frac{y^{7}}{7})}^{\frac{1}{5}})}^{5}$ .

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Step 3.3.2.1.1

Apply basic rules of exponents.

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Step 3.3.2.1.1.1

Apply the product rule to

$\frac{y^{7}}{7}$ .

${(4 \frac{{(y^{7})}^{\frac{1}{5}}}{7^{\frac{1}{5}}})}^{5} = x^{5}$

Step 3.3.2.1.1.2

Multiply the exponents in

${(y^{7})}^{\frac{1}{5}}$ .

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Step 3.3.2.1.1.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${(4 \frac{y^{7 (\frac{1}{5})}}{7^{\frac{1}{5}}})}^{5} = x^{5}$

Step 3.3.2.1.1.2.2

Combine

$7$ and

$\frac{1}{5}$ .

${(4 \frac{y^{\frac{7}{5}}}{7^{\frac{1}{5}}})}^{5} = x^{5}$

Step 3.3.2.1.2

Combine

$4$ and

$\frac{y^{\frac{7}{5}}}{7^{\frac{1}{5}}}$ .

${(\frac{4 y^{\frac{7}{5}}}{7^{\frac{1}{5}}})}^{5} = x^{5}$

Step 3.3.2.1.3

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.1.3.1

Apply the product rule to

$\frac{4 y^{\frac{7}{5}}}{7^{\frac{1}{5}}}$ .

$\frac{{(4 y^{\frac{7}{5}})}^{5}}{{(7^{\frac{1}{5}})}^{5}} = x^{5}$

Step 3.3.2.1.3.2

Apply the product rule to

$4 y^{\frac{7}{5}}$ .

$\frac{4^{5} {(y^{\frac{7}{5}})}^{5}}{{(7^{\frac{1}{5}})}^{5}} = x^{5}$

Step 3.3.2.1.4

Simplify the numerator.

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Step 3.3.2.1.4.1

Raise

$4$ to the power of

$5$ .

$\frac{1024 {(y^{\frac{7}{5}})}^{5}}{{(7^{\frac{1}{5}})}^{5}} = x^{5}$

Step 3.3.2.1.4.2

Multiply the exponents in

${(y^{\frac{7}{5}})}^{5}$ .

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Step 3.3.2.1.4.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{1024 y^{\frac{7}{5} \cdot 5}}{{(7^{\frac{1}{5}})}^{5}} = x^{5}$

Step 3.3.2.1.4.2.2

Cancel the common factor of

$5$ .

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Step 3.3.2.1.4.2.2.1

Cancel the common factor.

$\frac{1024 y^{\frac{7}{5} \cdot 5}}{{(7^{\frac{1}{5}})}^{5}} = x^{5}$

Step 3.3.2.1.4.2.2.2

Rewrite the expression.

$\frac{1024 y^{7}}{{(7^{\frac{1}{5}})}^{5}} = x^{5}$

Step 3.3.2.1.5

Simplify the denominator.

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Step 3.3.2.1.5.1

Multiply the exponents in

${(7^{\frac{1}{5}})}^{5}$ .

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Step 3.3.2.1.5.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{1024 y^{7}}{7^{\frac{1}{5} \cdot 5}} = x^{5}$

Step 3.3.2.1.5.1.2

Cancel the common factor of

$5$ .

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Step 3.3.2.1.5.1.2.1

Cancel the common factor.

$\frac{1024 y^{7}}{7^{\frac{1}{5} \cdot 5}} = x^{5}$

Step 3.3.2.1.5.1.2.2

Rewrite the expression.

$\frac{1024 y^{7}}{7^{1}} = x^{5}$

Step 3.3.2.1.5.2

Evaluate the exponent.

$\frac{1024 y^{7}}{7} = x^{5}$

Step 3.4

Solve for

$y$ .

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Step 3.4.1

Multiply both sides of the equation by

$\frac{7}{1024}$ .

$\frac{7}{1024} \cdot \frac{1024 y^{7}}{7} = \frac{7}{1024} x^{5}$

Step 3.4.2

Simplify both sides of the equation.

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Step 3.4.2.1

Simplify the left side.

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Step 3.4.2.1.1

Simplify

$\frac{7}{1024} \cdot \frac{1024 y^{7}}{7}$ .

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Step 3.4.2.1.1.1

Combine.

$\frac{7 (1024 y^{7})}{1024 \cdot 7} = \frac{7}{1024} x^{5}$

Step 3.4.2.1.1.2

Cancel the common factor of

$7$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

$\frac{7 (1024 y^{7})}{1024 \cdot 7} = \frac{7}{1024} x^{5}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

$\frac{1024 y^{7}}{1024} = \frac{7}{1024} x^{5}$

Step 3.4.2.1.1.3

Cancel the common factor of

$1024$ .

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Step 3.4.2.1.1.3.1

Cancel the common factor.

$\frac{1024 y^{7}}{1024} = \frac{7}{1024} x^{5}$

Step 3.4.2.1.1.3.2

Divide

$y^{7}$ by

$1$ .

$y^{7} = \frac{7}{1024} x^{5}$

Step 3.4.2.2

Simplify the right side.

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Step 3.4.2.2.1

Combine

$\frac{7}{1024}$ and

$x^{5}$ .

$y^{7} = \frac{7 x^{5}}{1024}$

Step 3.4.3

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

$y = \sqrt[7]{\frac{7 x^{5}}{1024}}$

Step 3.4.4

Simplify

$\sqrt[7]{\frac{7 x^{5}}{1024}}$ .

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Step 3.4.4.1

Rewrite

$\frac{7 x^{5}}{1024}$ as

${(\frac{1}{2})}^{7} \frac{7 x^{5}}{8}$ .

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Step 3.4.4.1.1

Factor the perfect power

$1^{7}$ out of

$7 x^{5}$ .

$y = \sqrt[7]{\frac{1^{7} (7 x^{5})}{1024}}$

Step 3.4.4.1.2

Factor the perfect power

$2^{7}$ out of

$1024$ .

$y = \sqrt[7]{\frac{1^{7} (7 x^{5})}{2^{7} \cdot 8}}$

Step 3.4.4.1.3

Rearrange the fraction

$\frac{1^{7} (7 x^{5})}{2^{7} \cdot 8}$ .

$y = \sqrt[7]{{(\frac{1}{2})}^{7} \frac{7 x^{5}}{8}}$

Step 3.4.4.2

Pull terms out from under the radical.

$y = \frac{1}{2} \sqrt[7]{\frac{7 x^{5}}{8}}$

Step 3.4.4.3

Rewrite

$\sqrt[7]{\frac{7 x^{5}}{8}}$ as

$\frac{\sqrt[7]{7 x^{5}}}{\sqrt[7]{8}}$ .

$y = \frac{1}{2} \cdot \frac{\sqrt[7]{7 x^{5}}}{\sqrt[7]{8}}$

Step 3.4.4.4

Combine.

$y = \frac{1 \sqrt[7]{7 x^{5}}}{2 \sqrt[7]{8}}$

Step 3.4.4.5

Multiply

$\sqrt[7]{7 x^{5}}$ by

$1$ .

$y = \frac{\sqrt[7]{7 x^{5}}}{2 \sqrt[7]{8}}$

Step 3.4.4.6

Multiply

$\frac{\sqrt[7]{7 x^{5}}}{2 \sqrt[7]{8}}$ by

$\frac{{\sqrt[7]{8}}^{6}}{{\sqrt[7]{8}}^{6}}$ .

$y = \frac{\sqrt[7]{7 x^{5}}}{2 \sqrt[7]{8}} \cdot \frac{{\sqrt[7]{8}}^{6}}{{\sqrt[7]{8}}^{6}}$

Step 3.4.4.7

Combine and simplify the denominator.

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Step 3.4.4.7.1

Multiply

$\frac{\sqrt[7]{7 x^{5}}}{2 \sqrt[7]{8}}$ by

$\frac{{\sqrt[7]{8}}^{6}}{{\sqrt[7]{8}}^{6}}$ .

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 \sqrt[7]{8} {\sqrt[7]{8}}^{6}}$

Step 3.4.4.7.2

Move

$\sqrt[7]{8}$ .

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 (\sqrt[7]{8} {\sqrt[7]{8}}^{6})}$

Step 3.4.4.7.3

Raise

$\sqrt[7]{8}$ to the power of

$1$ .

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 ({\sqrt[7]{8}}^{1} {\sqrt[7]{8}}^{6})}$

Step 3.4.4.7.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 {\sqrt[7]{8}}^{1 + 6}}$

Step 3.4.4.7.5

Add

$1$ and

$6$ .

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 {\sqrt[7]{8}}^{7}}$

Step 3.4.4.7.6

Rewrite

${\sqrt[7]{8}}^{7}$ as

$8$ .

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Step 3.4.4.7.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[7]{8}$ as

$8^{\frac{1}{7}}$ .

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 {(8^{\frac{1}{7}})}^{7}}$

Step 3.4.4.7.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 \cdot 8^{\frac{1}{7} \cdot 7}}$

Step 3.4.4.7.6.3

Combine

$\frac{1}{7}$ and

$7$ .

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 \cdot 8^{\frac{7}{7}}}$

Step 3.4.4.7.6.4

Cancel the common factor of

$7$ .

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Step 3.4.4.7.6.4.1

Cancel the common factor.

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 \cdot 8^{\frac{7}{7}}}$

Step 3.4.4.7.6.4.2

Rewrite the expression.

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 \cdot 8^{1}}$

Step 3.4.4.7.6.5

Evaluate the exponent.

$y = \frac{\sqrt[7]{7 x^{5}} {\sqrt[7]{8}}^{6}}{2 \cdot 8}$

Step 3.4.4.8

Simplify the numerator.

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Step 3.4.4.8.1

Rewrite

${\sqrt[7]{8}}^{6}$ as

$\sqrt[7]{8^{6}}$ .

$y = \frac{\sqrt[7]{7 x^{5}} \sqrt[7]{8^{6}}}{2 \cdot 8}$

Step 3.4.4.8.2

Raise

$8$ to the power of

$6$ .

$y = \frac{\sqrt[7]{7 x^{5}} \sqrt[7]{262144}}{2 \cdot 8}$

Step 3.4.4.8.3

Rewrite

$262144$ as

$4^{7} \cdot 16$ .

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Step 3.4.4.8.3.1

Factor

$16384$ out of

$262144$ .

$y = \frac{\sqrt[7]{7 x^{5}} \sqrt[7]{16384 (16)}}{2 \cdot 8}$

Step 3.4.4.8.3.2

Rewrite

$16384$ as

$4^{7}$ .

$y = \frac{\sqrt[7]{7 x^{5}} \sqrt[7]{4^{7} \cdot 16}}{2 \cdot 8}$

Step 3.4.4.8.4

Pull terms out from under the radical.

$y = \frac{\sqrt[7]{7 x^{5}} \cdot 4 \sqrt[7]{16}}{2 \cdot 8}$

Step 3.4.4.8.5

Combine exponents.

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Step 3.4.4.8.5.1

Combine using the product rule for radicals.

$y = \frac{4 \sqrt[7]{7 x^{5} \cdot 16}}{2 \cdot 8}$

Step 3.4.4.8.5.2

Multiply

$16$ by

$7$ .

$y = \frac{4 \sqrt[7]{112 x^{5}}}{2 \cdot 8}$

Step 3.4.4.9

Reduce the expression by cancelling the common factors.

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Step 3.4.4.9.1

Multiply

$2$ by

$8$ .

$y = \frac{4 \sqrt[7]{112 x^{5}}}{16}$

Step 3.4.4.9.2

Cancel the common factor of

$4$ and

$16$ .

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Step 3.4.4.9.2.1

Factor

$4$ out of

$4 \sqrt[7]{112 x^{5}}$ .

$y = \frac{4 (\sqrt[7]{112 x^{5}})}{16}$

Step 3.4.4.9.2.2

Cancel the common factors.

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Step 3.4.4.9.2.2.1

Factor

$4$ out of

$16$ .

$y = \frac{4 \sqrt[7]{112 x^{5}}}{4 \cdot 4}$

Step 3.4.4.9.2.2.2

Cancel the common factor.

$y = \frac{4 \sqrt[7]{112 x^{5}}}{4 \cdot 4}$

Step 3.4.4.9.2.2.3

Rewrite the expression.

$y = \frac{\sqrt[7]{112 x^{5}}}{4}$

Step 4

Replace

$y$ with

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

$f^{- 1} (x) = \frac{\sqrt[7]{112 x^{5}}}{4}$

Step 5

Verify if

$f^{- 1} (x) = \frac{\sqrt[7]{112 x^{5}}}{4}$ is the inverse of

$f (x) = 4 \sqrt[5]{\frac{x^{7}}{7}}$ .

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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

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}})$ by substituting in the value of

$f$ into

$f^{- 1}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 {(4 \sqrt[5]{\frac{x^{7}}{7}})}^{5}}}{4}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Apply the product rule to

$4 \sqrt[5]{\frac{x^{7}}{7}}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (4^{5} {\sqrt[5]{\frac{x^{7}}{7}}}^{5})}}{4}$

Step 5.2.3.2

Raise

$4$ to the power of

$5$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {\sqrt[5]{\frac{x^{7}}{7}}}^{5})}}{4}$

Step 5.2.3.3

Rewrite

$\sqrt[5]{\frac{x^{7}}{7}}$ as

$\frac{\sqrt[5]{x^{7}}}{\sqrt[5]{7}}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{\sqrt[5]{x^{7}}}{\sqrt[5]{7}})}^{5})}}{4}$

Step 5.2.3.4

Simplify the numerator.

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Step 5.2.3.4.1

Factor out

$x^{5}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{\sqrt[5]{x^{5} x^{2}}}{\sqrt[5]{7}})}^{5})}}{4}$

Step 5.2.3.4.2

Pull terms out from under the radical.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}}}{\sqrt[5]{7}})}^{5})}}{4}$

Step 5.2.3.5

Multiply

$\frac{x \sqrt[5]{x^{2}}}{\sqrt[5]{7}}$ by

$\frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}}}{\sqrt[5]{7}} \cdot \frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}})}^{5})}}{4}$

Step 5.2.3.6

Combine and simplify the denominator.

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Step 5.2.3.6.1

Multiply

$\frac{x \sqrt[5]{x^{2}}}{\sqrt[5]{7}}$ by

$\frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{\sqrt[5]{7} {\sqrt[5]{7}}^{4}})}^{5})}}{4}$

Step 5.2.3.6.2

Raise

$\sqrt[5]{7}$ to the power of

$1$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{\sqrt[5]{7} {\sqrt[5]{7}}^{4}})}^{5})}}{4}$

Step 5.2.3.6.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{1 + 4}})}^{5})}}{4}$

Step 5.2.3.6.4

Add

$1$ and

$4$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{5}})}^{5})}}{4}$

Step 5.2.3.6.5

Rewrite

${\sqrt[5]{7}}^{5}$ as

$7$ .

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Step 5.2.3.6.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[5]{7}$ as

$7^{\frac{1}{5}}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{{(7^{\frac{1}{5}})}^{5}})}^{5})}}{4}$

Step 5.2.3.6.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{7^{\frac{1}{5} \cdot 5}})}^{5})}}{4}$

Step 5.2.3.6.5.3

Combine

$\frac{1}{5}$ and

$5$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{7^{\frac{5}{5}}})}^{5})}}{4}$

Step 5.2.3.6.5.4

Cancel the common factor of

$5$ .

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Step 5.2.3.6.5.4.1

Cancel the common factor.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{7^{\frac{5}{5}}})}^{5})}}{4}$

Step 5.2.3.6.5.4.2

Rewrite the expression.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{7})}^{5})}}{4}$

Step 5.2.3.6.5.5

Evaluate the exponent.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} {\sqrt[5]{7}}^{4}}{7})}^{5})}}{4}$

Step 5.2.3.7

Simplify the numerator.

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Step 5.2.3.7.1

Rewrite

${\sqrt[5]{7}}^{4}$ as

$\sqrt[5]{7^{4}}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} \sqrt[5]{7^{4}}}{7})}^{5})}}{4}$

Step 5.2.3.7.2

Raise

$7$ to the power of

$4$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{x^{2}} \sqrt[5]{2401}}{7})}^{5})}}{4}$

Step 5.2.3.7.3

Combine using the product rule for radicals.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 {(\frac{x \sqrt[5]{2401 x^{2}}}{7})}^{5})}}{4}$

Step 5.2.3.8

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.3.8.1

Apply the product rule to

$\frac{x \sqrt[5]{2401 x^{2}}}{7}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{{(x \sqrt[5]{2401 x^{2}})}^{5}}{7^{5}}))}}{4}$

Step 5.2.3.8.2

Apply the product rule to

$x \sqrt[5]{2401 x^{2}}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{5} {\sqrt[5]{2401 x^{2}}}^{5}}{7^{5}}))}}{4}$

Step 5.2.3.9

Simplify the numerator.

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Step 5.2.3.9.1

Rewrite

${\sqrt[5]{2401 x^{2}}}^{5}$ as

$2401 x^{2}$ .

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Step 5.2.3.9.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[5]{2401 x^{2}}$ as

${(2401 x^{2})}^{\frac{1}{5}}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{5} {({(2401 x^{2})}^{\frac{1}{5}})}^{5}}{7^{5}}))}}{4}$

Step 5.2.3.9.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{5} {(2401 x^{2})}^{\frac{1}{5} \cdot 5}}{7^{5}}))}}{4}$

Step 5.2.3.9.1.3

Combine

$\frac{1}{5}$ and

$5$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{5} {(2401 x^{2})}^{\frac{5}{5}}}{7^{5}}))}}{4}$

Step 5.2.3.9.1.4

Cancel the common factor of

$5$ .

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Step 5.2.3.9.1.4.1

Cancel the common factor.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{5} {(2401 x^{2})}^{\frac{5}{5}}}{7^{5}}))}}{4}$

Step 5.2.3.9.1.4.2

Rewrite the expression.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{5} (2401 x^{2})}{7^{5}}))}}{4}$

Step 5.2.3.9.1.5

Simplify.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{5} (2401 x^{2})}{7^{5}}))}}{4}$

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{5} \cdot (2401 x^{2})}{7^{5}}))}}{4}$

Step 5.2.3.9.2

Multiply

$x^{5}$ by

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

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Step 5.2.3.9.2.1

Move

$x^{2}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{2} x^{5} \cdot 2401}{7^{5}}))}}{4}$

Step 5.2.3.9.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{2 + 5} \cdot 2401}{7^{5}}))}}{4}$

Step 5.2.3.9.2.3

Add

$2$ and

$5$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{7} \cdot 2401}{7^{5}}))}}{4}$

Step 5.2.3.10

Raise

$7$ to the power of

$5$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{7} \cdot 2401}{16807}))}}{4}$

Step 5.2.3.11

Cancel the common factor of

$2401$ and

$16807$ .

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Step 5.2.3.11.1

Factor

$2401$ out of

$x^{7} \cdot 2401$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{2401 \cdot x^{7}}{16807}))}}{4}$

Step 5.2.3.11.2

Cancel the common factors.

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Step 5.2.3.11.2.1

Factor

$2401$ out of

$16807$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{2401 \cdot x^{7}}{2401 \cdot 7}))}}{4}$

Step 5.2.3.11.2.2

Cancel the common factor.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{2401 \cdot x^{7}}{2401 \cdot 7}))}}{4}$

Step 5.2.3.11.2.3

Rewrite the expression.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{112 \cdot (1024 (\frac{x^{7}}{7}))}}{4}$

Step 5.2.3.12

Combine exponents.

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Step 5.2.3.12.1

Multiply

$112$ by

$1024$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{114688 (\frac{x^{7}}{7})}}{4}$

Step 5.2.3.12.2

Combine

$114688$ and

$\frac{x^{7}}{7}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{\frac{114688 x^{7}}{7}}}{4}$

Step 5.2.3.13

Reduce the expression by cancelling the common factors.

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Step 5.2.3.13.1

Reduce the expression

$\frac{114688 x^{7}}{7}$ by cancelling the common factors.

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Step 5.2.3.13.1.1

Factor

$7$ out of

$114688 x^{7}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{\frac{7 (16384 x^{7})}{7}}}{4}$

Step 5.2.3.13.1.2

Factor

$7$ out of

$7$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{\frac{7 (16384 x^{7})}{7 (1)}}}{4}$

Step 5.2.3.13.1.3

Cancel the common factor.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{\frac{7 (16384 x^{7})}{7 \cdot 1}}}{4}$

Step 5.2.3.13.1.4

Rewrite the expression.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{\frac{16384 x^{7}}{1}}}{4}$

Step 5.2.3.13.2

Divide

$16384 x^{7}$ by

$1$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{16384 x^{7}}}{4}$

Step 5.2.3.14

Rewrite

$16384 x^{7}$ as

${(4 x)}^{7}$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{\sqrt[7]{{(4 x)}^{7}}}{4}$

Step 5.2.3.15

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

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{4 x}{4}$

Step 5.2.4

Cancel the common factor of

$4$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = \frac{4 x}{4}$

Step 5.2.4.2

Divide

$x$ by

$1$ .

$f^{- 1} (4 \sqrt[5]{\frac{x^{7}}{7}}) = x$

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

$f (\frac{\sqrt[7]{112 x^{5}}}{4})$ by substituting in the value of

$f^{- 1}$ into

$f$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{(\frac{\sqrt[7]{112 x^{5}}}{4})}^{7}}{7}}$

Step 5.3.3

Apply the product rule to

$\frac{\sqrt[7]{112 x^{5}}}{4}$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{\frac{{\sqrt[7]{112 x^{5}}}^{7}}{4^{7}}}{7}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{\sqrt[7]{112 x^{5}}}^{7}}{4^{7}} \cdot \frac{1}{7}}$

Step 5.3.5

Combine.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{\sqrt[7]{112 x^{5}}}^{7} \cdot 1}{4^{7} \cdot 7}}$

Step 5.3.6

Simplify the expression.

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Step 5.3.6.1

Multiply

${\sqrt[7]{112 x^{5}}}^{7}$ by

$1$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{\sqrt[7]{112 x^{5}}}^{7}}{4^{7} \cdot 7}}$

Step 5.3.6.2

Raise

$4$ to the power of

$7$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{\sqrt[7]{112 x^{5}}}^{7}}{16384 \cdot 7}}$

Step 5.3.7

Rewrite

${\sqrt[7]{112 x^{5}}}^{7}$ as

$112 x^{5}$ .

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Step 5.3.7.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[7]{112 x^{5}}$ as

${(112 x^{5})}^{\frac{1}{7}}$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{({(112 x^{5})}^{\frac{1}{7}})}^{7}}{16384 \cdot 7}}$

Step 5.3.7.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{(112 x^{5})}^{\frac{1}{7} \cdot 7}}{16384 \cdot 7}}$

Step 5.3.7.3

Combine

$\frac{1}{7}$ and

$7$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{(112 x^{5})}^{\frac{7}{7}}}{16384 \cdot 7}}$

Step 5.3.7.4

Cancel the common factor of

$7$ .

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Step 5.3.7.4.1

Cancel the common factor.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{{(112 x^{5})}^{\frac{7}{7}}}{16384 \cdot 7}}$

Step 5.3.7.4.2

Rewrite the expression.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{112 x^{5}}{16384 \cdot 7}}$

Step 5.3.7.5

Simplify.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{112 x^{5}}{16384 \cdot 7}}$

Step 5.3.8

Multiply

$16384$ by

$7$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{112 x^{5}}{114688}}$

Step 5.3.9

Cancel the common factor of

$112$ and

$114688$ .

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Step 5.3.9.1

Factor

$112$ out of

$112 x^{5}$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{112 (x^{5})}{114688}}$

Step 5.3.9.2

Cancel the common factors.

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Step 5.3.9.2.1

Factor

$112$ out of

$114688$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{112 x^{5}}{112 \cdot 1024}}$

Step 5.3.9.2.2

Cancel the common factor.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{112 x^{5}}{112 \cdot 1024}}$

Step 5.3.9.2.3

Rewrite the expression.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{x^{5}}{1024}}$

Step 5.3.10

Rewrite

$1024$ as

$4^{5}$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{\frac{x^{5}}{4^{5}}}$

Step 5.3.11

Rewrite

$\frac{x^{5}}{4^{5}}$ as

${(\frac{x}{4})}^{5}$ .

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 \sqrt[5]{{(\frac{x}{4})}^{5}}$

Step 5.3.12

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

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 (\frac{x}{4})$

Step 5.3.13

Cancel the common factor of

$4$ .

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Step 5.3.13.1

Cancel the common factor.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = 4 (\frac{x}{4})$

Step 5.3.13.2

Rewrite the expression.

$f (\frac{\sqrt[7]{112 x^{5}}}{4}) = x$

Step 5.4

Since

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

$f (f^{- 1} (x)) = x$ , then

$f^{- 1} (x) = \frac{\sqrt[7]{112 x^{5}}}{4}$ is the inverse of

$f (x) = 4 \sqrt[5]{\frac{x^{7}}{7}}$ .

$f^{- 1} (x) = \frac{\sqrt[7]{112 x^{5}}}{4}$