Find the Inverse f(x) = seventh root of (x^3)/7

$f (x) = \sqrt[7]{\frac{x^{3}}{7}}$

Step 1

Write $f (x) = \sqrt[7]{\frac{x^{3}}{7}}$ as an equation.

$y = \sqrt[7]{\frac{x^{3}}{7}}$

Step 2

Interchange the variables.

$x = \sqrt[7]{\frac{y^{3}}{7}}$

Step 3

Solve for $y$ .

Rewrite the equation as $\sqrt[7]{\frac{y^{3}}{7}} = x$ .

$\sqrt[7]{\frac{y^{3}}{7}} = x$

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $7$ .

${\sqrt[7]{\frac{y^{3}}{7}}}^{7} = x^{7}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{\frac{y^{3}}{7}}$ as ${(\frac{y^{3}}{7})}^{\frac{1}{7}}$ .

${({(\frac{y^{3}}{7})}^{\frac{1}{7}})}^{7} = x^{7}$

Simplify the left side.

Simplify ${({(\frac{y^{3}}{7})}^{\frac{1}{7}})}^{7}$ .

Multiply the exponents in ${({(\frac{y^{3}}{7})}^{\frac{1}{7}})}^{7}$ .

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

${(\frac{y^{3}}{7})}^{\frac{1}{7} \cdot 7} = x^{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

${(\frac{y^{3}}{7})}^{\frac{1}{7} \cdot 7} = x^{7}$

Rewrite the expression.

${(\frac{y^{3}}{7})}^{1} = x^{7}$

Simplify.

$\frac{y^{3}}{7} = x^{7}$

Solve for $y$ .

Multiply both sides of the equation by $7$ .

$7 \frac{y^{3}}{7} = 7 x^{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$7 \frac{y^{3}}{7} = 7 x^{7}$

Rewrite the expression.

$y^{3} = 7 x^{7}$

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

$y = \sqrt[3]{7 x^{7}}$

Simplify $\sqrt[3]{7 x^{7}}$ .

Rewrite $7 x^{7}$ as ${(x^{2})}^{3} \cdot (7 x)$ .

Factor out $x^{6}$ .

$y = \sqrt[3]{7 (x^{6} x)}$

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$y = \sqrt[3]{7 ({(x^{2})}^{3} x)}$

Reorder $7$ and ${(x^{2})}^{3}$ .

$y = \sqrt[3]{{(x^{2})}^{3} \cdot 7 x}$

Add parentheses.

$y = \sqrt[3]{{(x^{2})}^{3} \cdot (7 x)}$

Pull terms out from under the radical.

$y = x^{2} \sqrt[3]{7 x}$

Step 4

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

$f^{- 1} (x) = x^{2} \sqrt[3]{7 x}$

Step 5

Verify if $f^{- 1} (x) = x^{2} \sqrt[3]{7 x}$ is the inverse of $f (x) = \sqrt[7]{\frac{x^{3}}{7}}$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = {(\sqrt[7]{\frac{x^{3}}{7}})}^{2} \sqrt[3]{7 (\sqrt[7]{\frac{x^{3}}{7}})}$

Rewrite $\sqrt[7]{\frac{x^{3}}{7}}$ as $\frac{\sqrt[7]{x^{3}}}{\sqrt[7]{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = {(\frac{\sqrt[7]{x^{3}}}{\sqrt[7]{7}})}^{2} \sqrt[3]{7 (\sqrt[7]{\frac{x^{3}}{7}})}$

Apply the product rule to $\frac{\sqrt[7]{x^{3}}}{\sqrt[7]{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{{\sqrt[7]{x^{3}}}^{2}}{{\sqrt[7]{7}}^{2}} \cdot \sqrt[3]{7 (\sqrt[7]{\frac{x^{3}}{7}})}$

Simplify the numerator.

Rewrite ${\sqrt[7]{x^{3}}}^{2}$ as ${({(x^{3})}^{2})}^{\frac{1}{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{{(x^{3})}^{2}}}{{\sqrt[7]{7}}^{2}} \cdot \sqrt[3]{7 (\sqrt[7]{\frac{x^{3}}{7}})}$

Multiply the exponents in ${(x^{3})}^{2}$ .

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

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{x^{3 \cdot 2}}}{{\sqrt[7]{7}}^{2}} \cdot \sqrt[3]{7 (\sqrt[7]{\frac{x^{3}}{7}})}$

Multiply $3$ by $2$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{x^{6}}}{{\sqrt[7]{7}}^{2}} \cdot \sqrt[3]{7 (\sqrt[7]{\frac{x^{3}}{7}})}$

Simplify the denominator.

Rewrite ${\sqrt[7]{7}}^{2}$ as ${(7^{2})}^{\frac{1}{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{x^{6}}}{\sqrt[7]{7^{2}}} \cdot \sqrt[3]{7 (\sqrt[7]{\frac{x^{3}}{7}})}$

Raise $7$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{x^{6}}}{\sqrt[7]{49}} \cdot \sqrt[3]{7 (\sqrt[7]{\frac{x^{3}}{7}})}$

Rewrite $\sqrt[7]{\frac{x^{3}}{7}}$ as $\frac{\sqrt[7]{x^{3}}}{\sqrt[7]{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{x^{6}}}{\sqrt[7]{49}} \cdot \sqrt[3]{7 (\frac{\sqrt[7]{x^{3}}}{\sqrt[7]{7}})}$

Combine $7$ and $\frac{\sqrt[7]{x^{3}}}{\sqrt[7]{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{x^{6}}}{\sqrt[7]{49}} \cdot \sqrt[3]{\frac{7 \sqrt[7]{x^{3}}}{\sqrt[7]{7}}}$

Rewrite $\sqrt[3]{\frac{7 \sqrt[7]{x^{3}}}{\sqrt[7]{7}}}$ as $\frac{\sqrt[3]{7 \sqrt[7]{x^{3}}}}{\sqrt[3]{\sqrt[7]{7}}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{x^{6}}}{\sqrt[7]{49}} \cdot \frac{\sqrt[3]{7 \sqrt[7]{x^{3}}}}{\sqrt[3]{\sqrt[7]{7}}}$

Combine.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[7]{x^{6}} \sqrt[3]{7 \sqrt[7]{x^{3}}}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Simplify the numerator.

Rewrite the expression using the least common index of $21$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{6}}$ as $x^{\frac{6}{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x^{\frac{6}{7}} \sqrt[3]{7 \sqrt[7]{x^{3}}}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Rewrite $x^{\frac{6}{7}}$ as $x^{\frac{18}{21}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x^{\frac{18}{21}} \sqrt[3]{7 \sqrt[7]{x^{3}}}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Rewrite $x^{\frac{18}{21}}$ as $\sqrt[21]{x^{18}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18}} \sqrt[3]{7 \sqrt[7]{x^{3}}}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7 \sqrt[7]{x^{3}}}$ as ${(7 \sqrt[7]{x^{3}})}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18}} {(7 \sqrt[7]{x^{3}})}^{\frac{1}{3}}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Rewrite ${(7 \sqrt[7]{x^{3}})}^{\frac{1}{3}}$ as ${(7 \sqrt[7]{x^{3}})}^{\frac{7}{21}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18}} {(7 \sqrt[7]{x^{3}})}^{\frac{7}{21}}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Rewrite ${(7 \sqrt[7]{x^{3}})}^{\frac{7}{21}}$ as $\sqrt[21]{{(7 \sqrt[7]{x^{3}})}^{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18}} \sqrt[21]{{(7 \sqrt[7]{x^{3}})}^{7}}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Combine using the product rule for radicals.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} {(7 \sqrt[7]{x^{3}})}^{7}}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Apply the product rule to $7 \sqrt[7]{x^{3}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (7^{7} {\sqrt[7]{x^{3}}}^{7})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Raise $7$ to the power of $7$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 {\sqrt[7]{x^{3}}}^{7})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Rewrite ${\sqrt[7]{x^{3}}}^{7}$ as $x^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{3}}$ as $x^{\frac{3}{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 {(x^{\frac{3}{7}})}^{7})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

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

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 x^{\frac{3}{7} \cdot 7})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Combine $\frac{3}{7}$ and $7$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 x^{\frac{3 \cdot 7}{7}})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Multiply $3$ by $7$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 x^{\frac{21}{7}})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Cancel the common factor of $21$ and $7$ .

Factor $7$ out of $21$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 x^{\frac{7 \cdot 3}{7}})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Cancel the common factors.

Factor $7$ out of $7$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 x^{\frac{7 \cdot 3}{7 (1)}})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Cancel the common factor.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 x^{\frac{7 \cdot 3}{7 \cdot 1}})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Rewrite the expression.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 x^{\frac{3}{1}})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Divide $3$ by $1$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{18} \cdot (823543 x^{3})}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Multiply $x^{18}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{3} x^{18} \cdot 823543}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

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

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{3 + 18} \cdot 823543}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Add $3$ and $18$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[7]{49} \sqrt[3]{\sqrt[7]{7}}}$

Simplify the denominator.

Rewrite the expression using the least common index of $21$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{49}$ as $49^{\frac{1}{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{49^{\frac{1}{7}} \sqrt[3]{\sqrt[7]{7}}}$

Rewrite $49^{\frac{1}{7}}$ as $49^{\frac{3}{21}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{49^{\frac{3}{21}} \sqrt[3]{\sqrt[7]{7}}}$

Rewrite $49^{\frac{3}{21}}$ as $\sqrt[21]{49^{3}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{49^{3}} \sqrt[3]{\sqrt[7]{7}}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\sqrt[7]{7}}$ as ${\sqrt[7]{7}}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{49^{3}} {\sqrt[7]{7}}^{\frac{1}{3}}}$

Rewrite ${\sqrt[7]{7}}^{\frac{1}{3}}$ as ${\sqrt[7]{7}}^{\frac{7}{21}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{49^{3}} {\sqrt[7]{7}}^{\frac{7}{21}}}$

Rewrite ${\sqrt[7]{7}}^{\frac{7}{21}}$ as $\sqrt[21]{{\sqrt[7]{7}}^{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{49^{3}} \sqrt[21]{{\sqrt[7]{7}}^{7}}}$

Combine using the product rule for radicals.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{49^{3} {\sqrt[7]{7}}^{7}}}$

Raise $49$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{117649 {\sqrt[7]{7}}^{7}}}$

Rewrite ${\sqrt[7]{7}}^{7}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{7}$ as $7^{\frac{1}{7}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{117649 {(7^{\frac{1}{7}})}^{7}}}$

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

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{117649 \cdot 7^{\frac{1}{7} \cdot 7}}}$

Combine $\frac{1}{7}$ and $7$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{117649 \cdot 7^{\frac{7}{7}}}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{117649 \cdot 7^{\frac{7}{7}}}}$

Rewrite the expression.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{117649 \cdot 7}}$

Evaluate the exponent.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{\sqrt[21]{x^{21} \cdot 823543}}{\sqrt[21]{117649 \cdot 7}}$

Simplify the numerator.

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[21]{823543}}{\sqrt[21]{117649 \cdot 7}}$

Rewrite $823543$ as $7^{7}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[21]{7^{7}}}{\sqrt[21]{117649 \cdot 7}}$

Rewrite $\sqrt[21]{7^{7}}$ as $\sqrt[3]{\sqrt[7]{7^{7}}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[3]{\sqrt[7]{7^{7}}}}{\sqrt[21]{117649 \cdot 7}}$

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

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[3]{7}}{\sqrt[21]{117649 \cdot 7}}$

Simplify the denominator.

Multiply $117649$ by $7$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[3]{7}}{\sqrt[21]{823543}}$

Rewrite $823543$ as $7^{7}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[3]{7}}{\sqrt[21]{7^{7}}}$

Rewrite $\sqrt[21]{7^{7}}$ as $\sqrt[3]{\sqrt[7]{7^{7}}}$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[3]{7}}{\sqrt[3]{\sqrt[7]{7^{7}}}}$

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

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[3]{7}}{\sqrt[3]{7}}$

Cancel the common factor of $\sqrt[3]{7}$ .

Cancel the common factor.

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = \frac{x \sqrt[3]{7}}{\sqrt[3]{7}}$

Divide $x$ by $1$ .

$f^{- 1} (\sqrt[7]{\frac{x^{3}}{7}}) = x$

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

Set up the composite result function.

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

Evaluate $f (x^{2} \sqrt[3]{7 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{{(x^{2} \sqrt[3]{7 x})}^{3}}{7}}$

Apply the product rule to $x^{2} \sqrt[3]{7 x}$ .

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{{(x^{2})}^{3} {\sqrt[3]{7 x}}^{3}}{7}}$

Simplify the numerator.

Multiply the exponents in ${(x^{2})}^{3}$ .

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

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{2 \cdot 3} {\sqrt[3]{7 x}}^{3}}{7}}$

Multiply $2$ by $3$ .

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{6} {\sqrt[3]{7 x}}^{3}}{7}}$

Rewrite ${\sqrt[3]{7 x}}^{3}$ as $7 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7 x}$ as ${(7 x)}^{\frac{1}{3}}$ .

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{6} {({(7 x)}^{\frac{1}{3}})}^{3}}{7}}$

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

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{6} {(7 x)}^{\frac{1}{3} \cdot 3}}{7}}$

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

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{6} {(7 x)}^{\frac{3}{3}}}{7}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{6} {(7 x)}^{\frac{3}{3}}}{7}}$

Rewrite the expression.

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{6} (7 x)}{7}}$

Simplify.

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{6} (7 x)}{7}}$

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{x^{6} \cdot (7 x)}{7}}$

Combine exponents.

Raise $x$ to the power of $1$ .

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{7 (x^{6} x)}{7}}$

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

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{7 x^{6 + 1}}{7}}$

Add $6$ and $1$ .

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{7 x^{7}}{7}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{\frac{7 x^{7}}{7}}$

Divide $x^{7}$ by $1$ .

$f (x^{2} \sqrt[3]{7 x}) = \sqrt[7]{x^{7}}$

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

$f (x^{2} \sqrt[3]{7 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{2} \sqrt[3]{7 x}$ is the inverse of $f (x) = \sqrt[7]{\frac{x^{3}}{7}}$ .

$f^{- 1} (x) = x^{2} \sqrt[3]{7 x}$

Find the Inverse f(x) = seventh root of (x^3)/7

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