Find the Inverse f(x)=(( seventh root of x)/5)^3

$f (x) = {(\frac{\sqrt[7]{x}}{5})}^{3}$

Write $f (x) = {(\frac{\sqrt[7]{x}}{5})}^{3}$ as an equation.

$y = {(\frac{\sqrt[7]{x}}{5})}^{3}$

Interchange the variables.

$x = {(\frac{\sqrt[7]{y}}{5})}^{3}$

Solve for $y$ .

Simplify ${(\frac{\sqrt[7]{y}}{5})}^{3}$ .

Apply the product rule to $\frac{\sqrt[7]{y}}{5}$ .

$x = \frac{{\sqrt[7]{y}}^{3}}{5^{3}}$

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

$x = \frac{\sqrt[7]{y^{3}}}{5^{3}}$

Raise $5$ to the power of $3$ .

$x = \frac{\sqrt[7]{y^{3}}}{125}$

Rewrite the equation as $\frac{\sqrt[7]{y^{3}}}{125} = x$ .

$\frac{\sqrt[7]{y^{3}}}{125} = x$

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

$\frac{y^{\frac{3}{7}}}{125} = x$

Multiply both sides of the equation by $125$ .

$125 \frac{y^{\frac{3}{7}}}{125} = 125 x$

Simplify the left side.

Cancel the common factor of $125$ .

Cancel the common factor.

$125 \frac{y^{\frac{3}{7}}}{125} = 125 x$

Rewrite the expression.

$y^{\frac{3}{7}} = 125 x$

Raise each side of the equation to the power of $\frac{7}{3}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{3}{7}})}^{\frac{7}{3}} = {(125 x)}^{\frac{7}{3}}$

Simplify the exponent.

Simplify the left side.

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

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

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

$y^{\frac{3}{7} \cdot \frac{7}{3}} = {(125 x)}^{\frac{7}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{3}{7} \cdot \frac{7}{3}} = {(125 x)}^{\frac{7}{3}}$

Rewrite the expression.

$y^{\frac{1}{7} \cdot 7} = {(125 x)}^{\frac{7}{3}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$y^{\frac{1}{7} \cdot 7} = {(125 x)}^{\frac{7}{3}}$

Rewrite the expression.

$y^{1} = {(125 x)}^{\frac{7}{3}}$

Simplify.

$y = {(125 x)}^{\frac{7}{3}}$

Simplify the right side.

Simplify ${(125 x)}^{\frac{7}{3}}$ .

Simplify the expression.

Apply the product rule to $125 x$ .

$y = 125^{\frac{7}{3}} x^{\frac{7}{3}}$

Rewrite $125$ as $5^{3}$ .

$y = {(5^{3})}^{\frac{7}{3}} x^{\frac{7}{3}}$

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

$y = 5^{3 (\frac{7}{3})} x^{\frac{7}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y = 5^{3 (\frac{7}{3})} x^{\frac{7}{3}}$

Rewrite the expression.

$y = 5^{7} x^{\frac{7}{3}}$

Raise $5$ to the power of $7$ .

$y = 78125 x^{\frac{7}{3}}$

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

$f^{- 1} (x) = 78125 x^{\frac{7}{3}}$

Verify if $f^{- 1} (x) = 78125 x^{\frac{7}{3}}$ is the inverse of $f (x) = {(\frac{\sqrt[7]{x}}{5})}^{3}$ .

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 $f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 {({(\frac{\sqrt[7]{x}}{5})}^{3})}^{\frac{7}{3}}$

Apply basic rules of exponents.

Multiply the exponents in ${({(\frac{\sqrt[7]{x}}{5})}^{3})}^{\frac{7}{3}}$ .

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 {(\frac{\sqrt[7]{x}}{5})}^{3 (\frac{7}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 {(\frac{\sqrt[7]{x}}{5})}^{3 (\frac{7}{3})}$

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 {(\frac{\sqrt[7]{x}}{5})}^{7}$

Apply the product rule to $\frac{\sqrt[7]{x}}{5}$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{{\sqrt[7]{x}}^{7}}{5^{7}})$

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

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{{(x^{\frac{1}{7}})}^{7}}{5^{7}})$

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{x^{\frac{1}{7} \cdot 7}}{5^{7}})$

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{x^{\frac{7}{7}}}{5^{7}})$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{x^{\frac{7}{7}}}{5^{7}})$

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{x}{5^{7}})$

Simplify.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{x}{5^{7}})$

Raise $5$ to the power of $7$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{x}{78125})$

Cancel the common factor of $78125$ .

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = 78125 (\frac{x}{78125})$

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{5})}^{3}) = x$

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

Set up the composite result function.

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

Evaluate $f (78125 x^{\frac{7}{3}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (78125 x^{\frac{7}{3}}) = {(\frac{\sqrt[7]{78125 x^{\frac{7}{3}}}}{5})}^{3}$

Simplify the numerator.

Rewrite $78125 x^{\frac{7}{3}}$ as ${(5 x^{\frac{1}{3}})}^{7}$ .

$f (78125 x^{\frac{7}{3}}) = {(\frac{\sqrt[7]{{(5 x^{\frac{1}{3}})}^{7}}}{5})}^{3}$

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

$f (78125 x^{\frac{7}{3}}) = {(\frac{5 x^{\frac{1}{3}}}{5})}^{3}$

Reduce the expression by cancelling the common factors.

Cancel the common factor.

$f (78125 x^{\frac{7}{3}}) = {(\frac{5 x^{\frac{1}{3}}}{5})}^{3}$

Simplify the expression.

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

$f (78125 x^{\frac{7}{3}}) = {(x^{\frac{1}{3}})}^{3}$

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

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

$f (78125 x^{\frac{7}{3}}) = x^{\frac{1}{3} \cdot 3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (78125 x^{\frac{7}{3}}) = x^{\frac{1}{3} \cdot 3}$

Rewrite the expression.

$f (78125 x^{\frac{7}{3}}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 78125 x^{\frac{7}{3}}$ is the inverse of $f (x) = {(\frac{\sqrt[7]{x}}{5})}^{3}$ .

$f^{- 1} (x) = 78125 x^{\frac{7}{3}}$

Find the Inverse f(x)=(( seventh root of x)/5)^3

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