Find the Inverse f(x)=((x^(1/7))/5)^5

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

Write $f (x) = {(\frac{x^{\frac{1}{7}}}{5})}^{5}$ as an equation.

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

Interchange the variables.

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

Solve for $y$ .

Rewrite the equation as ${(\frac{y^{\frac{1}{7}}}{5})}^{5} = x$ .

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

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

Apply the product rule to $\frac{y^{\frac{1}{7}}}{5}$ .

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

Multiply the exponents in ${(y^{\frac{1}{7}})}^{5}$ .

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

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

Combine $\frac{1}{7}$ and $5$ .

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

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

$\frac{y^{\frac{5}{7}}}{3125} = x$

Multiply both sides of the equation by $3125$ .

$3125 \frac{y^{\frac{5}{7}}}{3125} = 3125 x$

Simplify the left side.

Cancel the common factor of $3125$ .

Cancel the common factor.

$3125 \frac{y^{\frac{5}{7}}}{3125} = 3125 x$

Rewrite the expression.

$y^{\frac{5}{7}} = 3125 x$

Raise each side of the equation to the power of $\frac{7}{5}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{5}{7}})}^{\frac{7}{5}} = {(3125 x)}^{\frac{7}{5}}$

Simplify the exponent.

Simplify the left side.

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

Multiply the exponents in ${(y^{\frac{5}{7}})}^{\frac{7}{5}}$ .

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

$y^{\frac{5}{7} \cdot \frac{7}{5}} = {(3125 x)}^{\frac{7}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y^{\frac{5}{7} \cdot \frac{7}{5}} = {(3125 x)}^{\frac{7}{5}}$

Rewrite the expression.

$y^{\frac{1}{7} \cdot 7} = {(3125 x)}^{\frac{7}{5}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$y^{\frac{1}{7} \cdot 7} = {(3125 x)}^{\frac{7}{5}}$

Rewrite the expression.

$y^{1} = {(3125 x)}^{\frac{7}{5}}$

Simplify.

$y = {(3125 x)}^{\frac{7}{5}}$

Simplify the right side.

Simplify ${(3125 x)}^{\frac{7}{5}}$ .

Simplify the expression.

Apply the product rule to $3125 x$ .

$y = 3125^{\frac{7}{5}} x^{\frac{7}{5}}$

Rewrite $3125$ as $5^{5}$ .

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$y = 78125 x^{\frac{7}{5}}$

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

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

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

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

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

Apply basic rules of exponents.

Multiply the exponents in ${({(\frac{x^{\frac{1}{7}}}{5})}^{5})}^{\frac{7}{5}}$ .

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the product rule to $\frac{x^{\frac{1}{7}}}{5}$ .

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

Simplify the numerator.

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

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

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Reduce the expression by cancelling the common factors.

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

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

Cancel the common factor of $78125$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

Evaluate $f (78125 x^{\frac{7}{5}})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify the numerator.

Apply the product rule to $78125 x^{\frac{7}{5}}$ .

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

Rewrite $78125$ as $5^{7}$ .

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

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

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply the exponents in ${(x^{\frac{7}{5}})}^{\frac{1}{7}}$ .

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

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Reduce the expression by cancelling the common factors.

Cancel the common factor.

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

Simplify the expression.

Divide $x^{\frac{1}{5}}$ by $1$ .

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Find the Inverse f(x)=((x^(1/7))/5)^5

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