Find the Inverse f(x)=9((x^7)/4)^(1/5)

$f (x) = 9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}$

Write $f (x) = 9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}$ as an equation.

$y = 9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}$

Interchange the variables.

$x = 9 {(\frac{y^{7}}{4})}^{\frac{1}{5}}$

Solve for $y$ .

Rewrite the equation as $9 {(\frac{y^{7}}{4})}^{\frac{1}{5}} = x$ .

$9 {(\frac{y^{7}}{4})}^{\frac{1}{5}} = x$

Divide each term in $9 {(\frac{y^{7}}{4})}^{\frac{1}{5}} = x$ by $9$ and simplify.

Divide each term in $9 {(\frac{y^{7}}{4})}^{\frac{1}{5}} = x$ by $9$ .

$\frac{9 {(\frac{y^{7}}{4})}^{\frac{1}{5}}}{9} = \frac{x}{9}$

Simplify the left side.

Cancel the common factor.

$\frac{9 {(\frac{y^{7}}{4})}^{\frac{1}{5}}}{9} = \frac{x}{9}$

Simplify the expression.

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

${(\frac{y^{7}}{4})}^{\frac{1}{5}} = \frac{x}{9}$

Apply the product rule to $\frac{y^{7}}{4}$ .

$\frac{{(y^{7})}^{\frac{1}{5}}}{4^{\frac{1}{5}}} = \frac{x}{9}$

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

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

$\frac{y^{7 (\frac{1}{5})}}{4^{\frac{1}{5}}} = \frac{x}{9}$

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

$\frac{y^{\frac{7}{5}}}{4^{\frac{1}{5}}} = \frac{x}{9}$

Multiply both sides of the equation by $4^{\frac{1}{5}}$ .

$4^{\frac{1}{5}} \frac{y^{\frac{7}{5}}}{4^{\frac{1}{5}}} = 4^{\frac{1}{5}} \frac{x}{9}$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $4^{\frac{1}{5}}$ .

Cancel the common factor.

$4^{\frac{1}{5}} \frac{y^{\frac{7}{5}}}{4^{\frac{1}{5}}} = 4^{\frac{1}{5}} \frac{x}{9}$

Rewrite the expression.

$y^{\frac{7}{5}} = 4^{\frac{1}{5}} \frac{x}{9}$

Simplify the right side.

Simplify $4^{\frac{1}{5}} \frac{x}{9}$ .

Combine $4^{\frac{1}{5}}$ and $\frac{x}{9}$ .

$y^{\frac{7}{5}} = \frac{4^{\frac{1}{5}} x}{9}$

Reorder factors in $\frac{4^{\frac{1}{5}} x}{9}$ .

$y^{\frac{7}{5}} = \frac{x \cdot 4^{\frac{1}{5}}}{9}$

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

${(y^{\frac{7}{5}})}^{\frac{5}{7}} = {(\frac{x \cdot 4^{\frac{1}{5}}}{9})}^{\frac{5}{7}}$

Simplify the exponent.

Simplify the left side.

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

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

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

$y^{\frac{7}{5} \cdot \frac{5}{7}} = {(\frac{x \cdot 4^{\frac{1}{5}}}{9})}^{\frac{5}{7}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$y^{\frac{7}{5} \cdot \frac{5}{7}} = {(\frac{x \cdot 4^{\frac{1}{5}}}{9})}^{\frac{5}{7}}$

Rewrite the expression.

$y^{\frac{1}{5} \cdot 5} = {(\frac{x \cdot 4^{\frac{1}{5}}}{9})}^{\frac{5}{7}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y^{\frac{1}{5} \cdot 5} = {(\frac{x \cdot 4^{\frac{1}{5}}}{9})}^{\frac{5}{7}}$

Rewrite the expression.

$y^{1} = {(\frac{x \cdot 4^{\frac{1}{5}}}{9})}^{\frac{5}{7}}$

Simplify.

$y = {(\frac{x \cdot 4^{\frac{1}{5}}}{9})}^{\frac{5}{7}}$

Simplify the right side.

Simplify ${(\frac{x \cdot 4^{\frac{1}{5}}}{9})}^{\frac{5}{7}}$ .

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

Apply the product rule to $\frac{x \cdot 4^{\frac{1}{5}}}{9}$ .

$y = \frac{{(x \cdot 4^{\frac{1}{5}})}^{\frac{5}{7}}}{9^{\frac{5}{7}}}$

Apply the product rule to $x \cdot 4^{\frac{1}{5}}$ .

$y = \frac{x^{\frac{5}{7}} \cdot {(4^{\frac{1}{5}})}^{\frac{5}{7}}}{9^{\frac{5}{7}}}$

Multiply the exponents in ${(4^{\frac{1}{5}})}^{\frac{5}{7}}$ .

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

$y = \frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{5} \cdot \frac{5}{7}}}{9^{\frac{5}{7}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y = \frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{5} \cdot \frac{5}{7}}}{9^{\frac{5}{7}}}$

Rewrite the expression.

$y = \frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

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

$f^{- 1} (x) = \frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Verify if $f^{- 1} (x) = \frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$ is the inverse of $f (x) = 9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}$ .

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 $f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{{(9 {(\frac{x^{7}}{4})}^{\frac{1}{5}})}^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Simplify the numerator.

Apply the product rule to $\frac{x^{7}}{4}$ .

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{{(9 (\frac{{(x^{7})}^{\frac{1}{5}}}{4^{\frac{1}{5}}}))}^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

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

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

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{{(9 (\frac{x^{7 (\frac{1}{5})}}{4^{\frac{1}{5}}}))}^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

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

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{{(9 (\frac{x^{\frac{7}{5}}}{4^{\frac{1}{5}}}))}^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Combine $9$ and $\frac{x^{\frac{7}{5}}}{4^{\frac{1}{5}}}$ .

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{{(\frac{9 x^{\frac{7}{5}}}{4^{\frac{1}{5}}})}^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

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

Apply the product rule to $\frac{9 x^{\frac{7}{5}}}{4^{\frac{1}{5}}}$ .

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{{(9 x^{\frac{7}{5}})}^{\frac{5}{7}}}{{(4^{\frac{1}{5}})}^{\frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Apply the product rule to $9 x^{\frac{7}{5}}$ .

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} {(x^{\frac{7}{5}})}^{\frac{5}{7}}}{{(4^{\frac{1}{5}})}^{\frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Simplify the numerator.

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

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

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x^{\frac{7}{5} \cdot \frac{5}{7}}}{{(4^{\frac{1}{5}})}^{\frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x^{\frac{7}{5} \cdot \frac{5}{7}}}{{(4^{\frac{1}{5}})}^{\frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Rewrite the expression.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x^{\frac{1}{5} \cdot 5}}{{(4^{\frac{1}{5}})}^{\frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x^{\frac{1}{5} \cdot 5}}{{(4^{\frac{1}{5}})}^{\frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Rewrite the expression.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x}{{(4^{\frac{1}{5}})}^{\frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Simplify.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x}{{(4^{\frac{1}{5}})}^{\frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Multiply the exponents in ${(4^{\frac{1}{5}})}^{\frac{5}{7}}$ .

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

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x}{4^{\frac{1}{5} \cdot \frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x}{4^{\frac{1}{5} \cdot \frac{5}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Rewrite the expression.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x}{4^{\frac{1}{7}}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Combine $\frac{9^{\frac{5}{7}} x}{4^{\frac{1}{7}}}$ and $4^{\frac{1}{7}}$ .

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x \cdot 4^{\frac{1}{7}}}{4^{\frac{1}{7}}}}{9^{\frac{5}{7}}}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{9^{\frac{5}{7}} x \cdot 4^{\frac{1}{7}}}{4^{\frac{1}{7}}}$ by cancelling the common factors.

Cancel the common factor.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x \cdot 4^{\frac{1}{7}}}{4^{\frac{1}{7}}}}{9^{\frac{5}{7}}}$

Rewrite the expression.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{\frac{9^{\frac{5}{7}} x}{1}}{9^{\frac{5}{7}}}$

Divide $9^{\frac{5}{7}} x$ by $1$ .

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{9^{\frac{5}{7}} x}{9^{\frac{5}{7}}}$

Cancel the common factor.

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = \frac{9^{\frac{5}{7}} x}{9^{\frac{5}{7}}}$

Divide $x$ by $1$ .

$f^{- 1} (9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{{(\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}})}^{7}}{4})}^{\frac{1}{5}}$

Simplify the numerator.

Apply the product rule to $\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$ .

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{{(x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}})}^{7}}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

Simplify the numerator.

Apply the product rule to $x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}$ .

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{{(x^{\frac{5}{7}})}^{7} \cdot {(4^{\frac{1}{7}})}^{7}}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

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

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

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{\frac{5}{7} \cdot 7} \cdot {(4^{\frac{1}{7}})}^{7}}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{\frac{5}{7} \cdot 7} \cdot {(4^{\frac{1}{7}})}^{7}}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot {(4^{\frac{1}{7}})}^{7}}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

Multiply the exponents in ${(4^{\frac{1}{7}})}^{7}$ .

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

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot 4^{\frac{1}{7} \cdot 7}}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot 4^{\frac{1}{7} \cdot 7}}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot 4}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

Evaluate the exponent.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot 4}{{(9^{\frac{5}{7}})}^{7}}}{4})}^{\frac{1}{5}}$

Simplify the denominator.

Multiply the exponents in ${(9^{\frac{5}{7}})}^{7}$ .

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

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot 4}{9^{\frac{5}{7} \cdot 7}}}{4})}^{\frac{1}{5}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot 4}{9^{\frac{5}{7} \cdot 7}}}{4})}^{\frac{1}{5}}$

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot 4}{9^{5}}}{4})}^{\frac{1}{5}}$

Raise $9$ to the power of $5$ .

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{x^{5} \cdot 4}{59049}}{4})}^{\frac{1}{5}}$

Move $4$ to the left of $x^{5}$ .

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{\frac{4 x^{5}}{59049}}{4})}^{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{4 x^{5}}{59049} \cdot \frac{1}{4})}^{\frac{1}{5}}$

Simplify terms.

Combine.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{4 x^{5} \cdot 1}{59049 \cdot 4})}^{\frac{1}{5}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{4 x^{5} \cdot 1}{59049 \cdot 4})}^{\frac{1}{5}}$

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{x^{5} \cdot 1}{59049})}^{\frac{1}{5}}$

Simplify the expression.

Multiply $x^{5}$ by $1$ .

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 {(\frac{x^{5}}{59049})}^{\frac{1}{5}}$

Apply the product rule to $\frac{x^{5}}{59049}$ .

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{{(x^{5})}^{\frac{1}{5}}}{59049^{\frac{1}{5}}})$

Simplify the numerator.

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

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

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x^{5 (\frac{1}{5})}}{59049^{\frac{1}{5}}})$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x^{5 (\frac{1}{5})}}{59049^{\frac{1}{5}}})$

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x}{59049^{\frac{1}{5}}})$

Simplify.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x}{59049^{\frac{1}{5}}})$

Simplify the denominator.

Rewrite $59049$ as $9^{5}$ .

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x}{{(9^{5})}^{\frac{1}{5}}})$

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

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x}{9^{5 (\frac{1}{5})}})$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x}{9^{5 (\frac{1}{5})}})$

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x}{9})$

Evaluate the exponent.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x}{9})$

Cancel the common factor of $9$ .

Cancel the common factor.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = 9 (\frac{x}{9})$

Rewrite the expression.

$f (\frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$ is the inverse of $f (x) = 9 {(\frac{x^{7}}{4})}^{\frac{1}{5}}$ .

$f^{- 1} (x) = \frac{x^{\frac{5}{7}} \cdot 4^{\frac{1}{7}}}{9^{\frac{5}{7}}}$

Find the Inverse f(x)=9((x^7)/4)^(1/5)

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