Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Simplify the expression.

Tap for more steps...

Step 3.2.2.2.1

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

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

Step 3.2.2.2.2

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

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

Step 3.2.2.2.3

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

Tap for more steps...

Step 3.2.2.2.3.1

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

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

Step 3.2.2.2.3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify both sides of the equation.

Tap for more steps...

Step 3.4.1

Simplify the left side.

Tap for more steps...

Step 3.4.1.1

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

Tap for more steps...

Step 3.4.1.1.1

Cancel the common factor.

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

Step 3.4.1.1.2

Rewrite the expression.

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

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

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

Tap for more steps...

Step 3.4.2.1.1

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

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

Step 3.4.2.1.2

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

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

Step 3.5

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

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

Step 3.6

Simplify the exponent.

Tap for more steps...

Step 3.6.1

Simplify the left side.

Tap for more steps...

Step 3.6.1.1

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

Tap for more steps...

Step 3.6.1.1.1

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

Tap for more steps...

Step 3.6.1.1.1.1

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

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

Step 3.6.1.1.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 3.6.1.1.1.2.1

Cancel the common factor.

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

Step 3.6.1.1.1.2.2

Rewrite the expression.

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

Step 3.6.1.1.1.3

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.6.1.1.1.3.1

Cancel the common factor.

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

Step 3.6.1.1.1.3.2

Rewrite the expression.

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

Step 3.6.1.1.2

Simplify.

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

Step 3.6.2

Simplify the right side.

Tap for more steps...

Step 3.6.2.1

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

Tap for more steps...

Step 3.6.2.1.1

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

Tap for more steps...

Step 3.6.2.1.1.1

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

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

Step 3.6.2.1.1.2

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

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

Step 3.6.2.1.2

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

Tap for more steps...

Step 3.6.2.1.2.1

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

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

Step 3.6.2.1.2.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.6.2.1.2.2.1

Cancel the common factor.

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

Step 3.6.2.1.2.2.2

Rewrite the expression.

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

Step 4

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

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

Step 5

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

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1

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

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

Step 5.2.3.2

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

Tap for more steps...

Step 5.2.3.2.1

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

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

Step 5.2.3.2.2

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

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

Step 5.2.3.3

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

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

Step 5.2.3.4

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

Tap for more steps...

Step 5.2.3.4.1

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

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

Step 5.2.3.4.2

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

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

Step 5.2.3.5

Simplify the numerator.

Tap for more steps...

Step 5.2.3.5.1

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

Tap for more steps...

Step 5.2.3.5.1.1

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

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

Step 5.2.3.5.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.2.3.5.1.2.1

Cancel the common factor.

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

Step 5.2.3.5.1.2.2

Rewrite the expression.

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

Step 5.2.3.5.1.3

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.3.5.1.3.1

Cancel the common factor.

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

Step 5.2.3.5.1.3.2

Rewrite the expression.

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

Step 5.2.3.5.2

Simplify.

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

Step 5.2.3.6

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

Tap for more steps...

Step 5.2.3.6.1

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

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

Step 5.2.3.6.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.3.6.2.1

Cancel the common factor.

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

Step 5.2.3.6.2.2

Rewrite the expression.

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

Step 5.2.4

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

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

Step 5.2.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.5.1

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

Tap for more steps...

Step 5.2.5.1.1

Cancel the common factor.

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

Step 5.2.5.1.2

Rewrite the expression.

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

Step 5.2.5.2

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

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

Step 5.2.6

Cancel the common factor.

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

Step 5.2.7

Divide $x$ by $1$ .

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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 $f$ .

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

Step 5.3.3

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1

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

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

Step 5.3.3.2

Simplify the numerator.

Tap for more steps...

Step 5.3.3.2.1

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

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

Step 5.3.3.2.2

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

Tap for more steps...

Step 5.3.3.2.2.1

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

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

Step 5.3.3.2.2.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.3.3.2.2.2.1

Cancel the common factor.

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

Step 5.3.3.2.2.2.2

Rewrite the expression.

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

Step 5.3.3.2.3

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

Tap for more steps...

Step 5.3.3.2.3.1

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

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

Step 5.3.3.2.3.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.3.3.2.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.3.2.2

Rewrite the expression.

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

Step 5.3.3.2.4

Evaluate the exponent.

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

Step 5.3.3.3

Simplify the denominator.

Tap for more steps...

Step 5.3.3.3.1

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

Tap for more steps...

Step 5.3.3.3.1.1

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

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

Step 5.3.3.3.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.3.3.3.1.2.1

Cancel the common factor.

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

Step 5.3.3.3.1.2.2

Rewrite the expression.

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

Step 5.3.3.3.2

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

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

Step 5.3.3.4

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

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

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.5

Simplify terms.

Tap for more steps...

Step 5.3.5.1

Combine.

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

Step 5.3.5.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.3.5.2.1

Cancel the common factor.

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

Step 5.3.5.2.2

Rewrite the expression.

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

Step 5.3.5.3

Simplify the expression.

Tap for more steps...

Step 5.3.5.3.1

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

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

Step 5.3.5.3.2

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

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

Step 5.3.6

Simplify the numerator.

Tap for more steps...

Step 5.3.6.1

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

Tap for more steps...

Step 5.3.6.1.1

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

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

Step 5.3.6.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.6.1.2.1

Cancel the common factor.

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

Step 5.3.6.1.2.2

Rewrite the expression.

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

Step 5.3.6.2

Simplify.

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

Step 5.3.7

Simplify the denominator.

Tap for more steps...

Step 5.3.7.1

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

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

Step 5.3.7.2

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

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

Step 5.3.7.3

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.7.3.1

Cancel the common factor.

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

Step 5.3.7.3.2

Rewrite the expression.

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

Step 5.3.7.4

Evaluate the exponent.

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

Step 5.3.8

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.3.8.1

Cancel the common factor.

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

Step 5.3.8.2

Rewrite the expression.

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

Step 5.4

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

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