Find the Inverse f(x)=7((x^(1/3))/6+5)

$f (x) = 7 (\frac{x^{\frac{1}{3}}}{6} + 5)$

Write $f (x) = 7 (\frac{x^{\frac{1}{3}}}{6} + 5)$ as an equation.

$y = 7 (\frac{x^{\frac{1}{3}}}{6} + 5)$

Interchange the variables.

$x = 7 (\frac{y^{\frac{1}{3}}}{6} + 5)$

Solve for $y$ .

Rewrite the equation as $7 (\frac{y^{\frac{1}{3}}}{6} + 5) = x$ .

$7 (\frac{y^{\frac{1}{3}}}{6} + 5) = x$

Divide each term in $7 (\frac{y^{\frac{1}{3}}}{6} + 5) = x$ by $7$ and simplify.

Divide each term in $7 (\frac{y^{\frac{1}{3}}}{6} + 5) = x$ by $7$ .

$\frac{7 (\frac{y^{\frac{1}{3}}}{6} + 5)}{7} = \frac{x}{7}$

Simplify the left side.

Cancel the common factor.

$\frac{7 (\frac{y^{\frac{1}{3}}}{6} + 5)}{7} = \frac{x}{7}$

Divide $\frac{y^{\frac{1}{3}}}{6} + 5$ by $1$ .

$\frac{y^{\frac{1}{3}}}{6} + 5 = \frac{x}{7}$

Move all terms not containing $y$ to the right side of the equation.

Subtract $5$ from both sides of the equation.

$\frac{y^{\frac{1}{3}}}{6} - \frac{x}{7} = - 5$

Add $\frac{x}{7}$ to both sides of the equation.

$\frac{y^{\frac{1}{3}}}{6} = - 5 + \frac{x}{7}$

Multiply both sides of the equation by $6$ .

$6 \frac{y^{\frac{1}{3}}}{6} = 6 (- 5 + \frac{x}{7})$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$6 \frac{y^{\frac{1}{3}}}{6} = 6 (- 5 + \frac{x}{7})$

Rewrite the expression.

$y^{\frac{1}{3}} = 6 (- 5 + \frac{x}{7})$

Simplify the right side.

Simplify $6 (- 5 + \frac{x}{7})$ .

Apply the distributive property.

$y^{\frac{1}{3}} = 6 \cdot - 5 + 6 \frac{x}{7}$

Multiply $6$ by $- 5$ .

$y^{\frac{1}{3}} = - 30 + 6 \frac{x}{7}$

Combine $6$ and $\frac{x}{7}$ .

$y^{\frac{1}{3}} = - 30 + \frac{6 x}{7}$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{3}})}^{3} = {(- 30 + \frac{6 x}{7})}^{3}$

Simplify the exponent.

Simplify the left side.

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

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

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

$y^{\frac{1}{3} \cdot 3} = {(- 30 + \frac{6 x}{7})}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(- 30 + \frac{6 x}{7})}^{3}$

Rewrite the expression.

$y^{1} = {(- 30 + \frac{6 x}{7})}^{3}$

Simplify.

$y = {(- 30 + \frac{6 x}{7})}^{3}$

Simplify the right side.

Simplify ${(- 30 + \frac{6 x}{7})}^{3}$ .

Use the Binomial Theorem.

$y = {(- 30)}^{3} + 3 {(- 30)}^{2} \frac{6 x}{7} + 3 \cdot - 30 {(\frac{6 x}{7})}^{2} + {(\frac{6 x}{7})}^{3}$

Simplify each term.

Raise $- 30$ to the power of $3$ .

$y = - 27000 + 3 {(- 30)}^{2} \frac{6 x}{7} + 3 \cdot - 30 {(\frac{6 x}{7})}^{2} + {(\frac{6 x}{7})}^{3}$

Raise $- 30$ to the power of $2$ .

$y = - 27000 + 3 \cdot 900 \frac{6 x}{7} + 3 \cdot - 30 {(\frac{6 x}{7})}^{2} + {(\frac{6 x}{7})}^{3}$

Multiply $3$ by $900$ .

$y = - 27000 + 2700 \frac{6 x}{7} + 3 \cdot - 30 {(\frac{6 x}{7})}^{2} + {(\frac{6 x}{7})}^{3}$

Multiply $2700 \frac{6 x}{7}$ .

Combine $2700$ and $\frac{6 x}{7}$ .

$y = - 27000 + \frac{2700 (6 x)}{7} + 3 \cdot - 30 {(\frac{6 x}{7})}^{2} + {(\frac{6 x}{7})}^{3}$

Multiply $6$ by $2700$ .

$y = - 27000 + \frac{16200 x}{7} + 3 \cdot - 30 {(\frac{6 x}{7})}^{2} + {(\frac{6 x}{7})}^{3}$

Multiply $3$ by $- 30$ .

$y = - 27000 + \frac{16200 x}{7} - 90 {(\frac{6 x}{7})}^{2} + {(\frac{6 x}{7})}^{3}$

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

Apply the product rule to $\frac{6 x}{7}$ .

$y = - 27000 + \frac{16200 x}{7} - 90 \frac{{(6 x)}^{2}}{7^{2}} + {(\frac{6 x}{7})}^{3}$

Apply the product rule to $6 x$ .

$y = - 27000 + \frac{16200 x}{7} - 90 \frac{6^{2} x^{2}}{7^{2}} + {(\frac{6 x}{7})}^{3}$

Raise $6$ to the power of $2$ .

$y = - 27000 + \frac{16200 x}{7} - 90 \frac{36 x^{2}}{7^{2}} + {(\frac{6 x}{7})}^{3}$

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

$y = - 27000 + \frac{16200 x}{7} - 90 \frac{36 x^{2}}{49} + {(\frac{6 x}{7})}^{3}$

Multiply $- 90 \frac{36 x^{2}}{49}$ .

Combine $- 90$ and $\frac{36 x^{2}}{49}$ .

$y = - 27000 + \frac{16200 x}{7} + \frac{- 90 (36 x^{2})}{49} + {(\frac{6 x}{7})}^{3}$

Multiply $36$ by $- 90$ .

$y = - 27000 + \frac{16200 x}{7} + \frac{- 3240 x^{2}}{49} + {(\frac{6 x}{7})}^{3}$

Move the negative in front of the fraction.

$y = - 27000 + \frac{16200 x}{7} - \frac{(3240) x^{2}}{49} + {(\frac{6 x}{7})}^{3}$

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

Apply the product rule to $\frac{6 x}{7}$ .

$y = - 27000 + \frac{16200 x}{7} - \frac{3240 x^{2}}{49} + \frac{{(6 x)}^{3}}{7^{3}}$

Apply the product rule to $6 x$ .

$y = - 27000 + \frac{16200 x}{7} - \frac{3240 x^{2}}{49} + \frac{6^{3} x^{3}}{7^{3}}$

Raise $6$ to the power of $3$ .

$y = - 27000 + \frac{16200 x}{7} - \frac{3240 x^{2}}{49} + \frac{216 x^{3}}{7^{3}}$

Raise $7$ to the power of $3$ .

$y = - 27000 + \frac{16200 x}{7} - \frac{3240 x^{2}}{49} + \frac{216 x^{3}}{343}$

Simplify $- 27000 + \frac{16200 x}{7} - \frac{3240 x^{2}}{49} + \frac{216 x^{3}}{343}$ .

Move $- 27000$ .

$y = \frac{16200 x}{7} - \frac{3240 x^{2}}{49} + \frac{216 x^{3}}{343} - 27000$

Move $\frac{16200 x}{7}$ .

$y = - \frac{3240 x^{2}}{49} + \frac{216 x^{3}}{343} + \frac{16200 x}{7} - 27000$

Reorder $- \frac{3240 x^{2}}{49}$ and $\frac{216 x^{3}}{343}$ .

$y = \frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000$

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

$f^{- 1} (x) = \frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000$

Verify if $f^{- 1} (x) = \frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000$ is the inverse of $f (x) = 7 (\frac{x^{\frac{1}{3}}}{6} + 5)$ .

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 $f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{216 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{3}}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Simplify each term.

Simplify the numerator.

Apply the product rule to $7 (\frac{x^{\frac{1}{3}}}{6} + 5)$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{216 \cdot (7^{3} {(\frac{x^{\frac{1}{3}}}{6} + 5)}^{3})}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Raise $7$ to the power of $3$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{216 \cdot (343 {(\frac{x^{\frac{1}{3}}}{6} + 5)}^{3})}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

To write $5$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{216 \cdot (343 {(\frac{x^{\frac{1}{3}}}{6} + 5 \cdot \frac{6}{6})}^{3})}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Combine $5$ and $\frac{6}{6}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{216 \cdot (343 {(\frac{x^{\frac{1}{3}}}{6} + \frac{5 \cdot 6}{6})}^{3})}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Combine the numerators over the common denominator.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{216 \cdot (343 {(\frac{x^{\frac{1}{3}} + 5 \cdot 6}{6})}^{3})}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $5$ by $6$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{216 \cdot (343 {(\frac{x^{\frac{1}{3}} + 30}{6})}^{3})}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $216$ by $343$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{74088 {(\frac{x^{\frac{1}{3}} + 30}{6})}^{3}}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Apply the product rule to $\frac{x^{\frac{1}{3}} + 30}{6}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{74088 (\frac{{(x^{\frac{1}{3}} + 30)}^{3}}{6^{3}})}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Raise $6$ to the power of $3$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{74088 (\frac{{(x^{\frac{1}{3}} + 30)}^{3}}{216})}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Combine $74088$ and $\frac{{(x^{\frac{1}{3}} + 30)}^{3}}{216}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{\frac{74088 {(x^{\frac{1}{3}} + 30)}^{3}}{216}}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{74088 {(x^{\frac{1}{3}} + 30)}^{3}}{216}$ by cancelling the common factors.

Factor $216$ out of $74088 {(x^{\frac{1}{3}} + 30)}^{3}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{\frac{216 (343 {(x^{\frac{1}{3}} + 30)}^{3})}{216}}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Factor $216$ out of $216$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{\frac{216 (343 {(x^{\frac{1}{3}} + 30)}^{3})}{216 (1)}}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Cancel the common factor.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{\frac{216 (343 {(x^{\frac{1}{3}} + 30)}^{3})}{216 \cdot 1}}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Rewrite the expression.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{\frac{343 {(x^{\frac{1}{3}} + 30)}^{3}}{1}}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Divide $343 {(x^{\frac{1}{3}} + 30)}^{3}$ by $1$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = \frac{343 {(x^{\frac{1}{3}} + 30)}^{3}}{343} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Cancel the common factor.

Divide ${(x^{\frac{1}{3}} + 30)}^{3}$ by $1$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = {(x^{\frac{1}{3}} + 30)}^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Use the Binomial Theorem.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = {(x^{\frac{1}{3}})}^{3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 30 + 3 x^{\frac{1}{3}} \cdot 30^{2} + 30^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Simplify each term.

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

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

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x^{\frac{1}{3} \cdot 3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 30 + 3 x^{\frac{1}{3}} \cdot 30^{2} + 30^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Cancel the common factor of $3$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot 30 + 3 x^{\frac{1}{3}} \cdot 30^{2} + 30^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Simplify.

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 3 x^{\frac{1}{3} \cdot 2} \cdot 30 + 3 x^{\frac{1}{3}} \cdot 30^{2} + 30^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Combine $\frac{1}{3}$ and $2$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 3 x^{\frac{2}{3}} \cdot 30 + 3 x^{\frac{1}{3}} \cdot 30^{2} + 30^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $30$ by $3$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 30^{2} + 30^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Raise $30$ to the power of $2$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 900 + 30^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $900$ by $3$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 30^{3} - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Raise $30$ to the power of $3$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{3240 {(7 (\frac{x^{\frac{1}{3}}}{6} + 5))}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Simplify the numerator.

Apply the product rule to $7 (\frac{x^{\frac{1}{3}}}{6} + 5)$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{3240 \cdot (7^{2} {(\frac{x^{\frac{1}{3}}}{6} + 5)}^{2})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

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

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{3240 \cdot (49 {(\frac{x^{\frac{1}{3}}}{6} + 5)}^{2})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

To write $5$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{3240 \cdot (49 {(\frac{x^{\frac{1}{3}}}{6} + 5 \cdot \frac{6}{6})}^{2})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Combine $5$ and $\frac{6}{6}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{3240 \cdot (49 {(\frac{x^{\frac{1}{3}}}{6} + \frac{5 \cdot 6}{6})}^{2})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Combine the numerators over the common denominator.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{3240 \cdot (49 {(\frac{x^{\frac{1}{3}} + 5 \cdot 6}{6})}^{2})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $5$ by $6$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{3240 \cdot (49 {(\frac{x^{\frac{1}{3}} + 30}{6})}^{2})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $3240$ by $49$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{158760 {(\frac{x^{\frac{1}{3}} + 30}{6})}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Apply the product rule to $\frac{x^{\frac{1}{3}} + 30}{6}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{158760 (\frac{{(x^{\frac{1}{3}} + 30)}^{2}}{6^{2}})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Raise $6$ to the power of $2$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{158760 (\frac{{(x^{\frac{1}{3}} + 30)}^{2}}{36})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Combine $158760$ and $\frac{{(x^{\frac{1}{3}} + 30)}^{2}}{36}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{\frac{158760 {(x^{\frac{1}{3}} + 30)}^{2}}{36}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{158760 {(x^{\frac{1}{3}} + 30)}^{2}}{36}$ by cancelling the common factors.

Factor $36$ out of $158760 {(x^{\frac{1}{3}} + 30)}^{2}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{\frac{36 (4410 {(x^{\frac{1}{3}} + 30)}^{2})}{36}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Factor $36$ out of $36$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{\frac{36 (4410 {(x^{\frac{1}{3}} + 30)}^{2})}{36 (1)}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Cancel the common factor.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{\frac{36 (4410 {(x^{\frac{1}{3}} + 30)}^{2})}{36 \cdot 1}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Rewrite the expression.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{\frac{4410 {(x^{\frac{1}{3}} + 30)}^{2}}{1}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Divide $4410 {(x^{\frac{1}{3}} + 30)}^{2}$ by $1$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{4410 {(x^{\frac{1}{3}} + 30)}^{2}}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Factor $49$ out of $4410 {(x^{\frac{1}{3}} + 30)}^{2}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{49 (90 {(x^{\frac{1}{3}} + 30)}^{2})}{49} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Cancel the common factors.

Factor $49$ out of $49$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{49 (90 {(x^{\frac{1}{3}} + 30)}^{2})}{49 (1)} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Cancel the common factor.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{49 (90 {(x^{\frac{1}{3}} + 30)}^{2})}{49 \cdot 1} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Rewrite the expression.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - \frac{90 {(x^{\frac{1}{3}} + 30)}^{2}}{1} + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Divide $90 {(x^{\frac{1}{3}} + 30)}^{2}$ by $1$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 {(x^{\frac{1}{3}} + 30)}^{2}) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Rewrite ${(x^{\frac{1}{3}} + 30)}^{2}$ as $(x^{\frac{1}{3}} + 30) (x^{\frac{1}{3}} + 30)$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 ((x^{\frac{1}{3}} + 30) (x^{\frac{1}{3}} + 30))) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Expand $(x^{\frac{1}{3}} + 30) (x^{\frac{1}{3}} + 30)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{1}{3}} (x^{\frac{1}{3}} + 30) + 30 (x^{\frac{1}{3}} + 30))) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Apply the distributive property.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot 30 + 30 (x^{\frac{1}{3}} + 30))) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Apply the distributive property.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot 30 + 30 x^{\frac{1}{3}} + 30 \cdot 30)) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Simplify and combine like terms.

Simplify each term.

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{1}{3} + \frac{1}{3}} + x^{\frac{1}{3}} \cdot 30 + 30 x^{\frac{1}{3}} + 30 \cdot 30)) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Combine the numerators over the common denominator.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{1 + 1}{3}} + x^{\frac{1}{3}} \cdot 30 + 30 x^{\frac{1}{3}} + 30 \cdot 30)) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Add $1$ and $1$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot 30 + 30 x^{\frac{1}{3}} + 30 \cdot 30)) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Move $30$ to the left of $x^{\frac{1}{3}}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{2}{3}} + 30 \cdot x^{\frac{1}{3}} + 30 x^{\frac{1}{3}} + 30 \cdot 30)) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $30$ by $30$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{2}{3}} + 30 x^{\frac{1}{3}} + 30 x^{\frac{1}{3}} + 900)) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Add $30 x^{\frac{1}{3}}$ and $30 x^{\frac{1}{3}}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 (x^{\frac{2}{3}} + 60 x^{\frac{1}{3}} + 900)) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Apply the distributive property.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 x^{\frac{2}{3}} + 90 (60 x^{\frac{1}{3}}) + 90 \cdot 900) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Simplify.

Multiply $60$ by $90$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 x^{\frac{2}{3}} + 5400 x^{\frac{1}{3}} + 90 \cdot 900) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $90$ by $900$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 x^{\frac{2}{3}} + 5400 x^{\frac{1}{3}} + 81000) + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Apply the distributive property.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - (90 x^{\frac{2}{3}}) - (5400 x^{\frac{1}{3}}) - 1 \cdot 81000 + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Simplify.

Multiply $90$ by $- 1$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - (5400 x^{\frac{1}{3}}) - 1 \cdot 81000 + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $5400$ by $- 1$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - 5400 x^{\frac{1}{3}} - 1 \cdot 81000 + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Multiply $- 1$ by $81000$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - 5400 x^{\frac{1}{3}} - 81000 + \frac{16200 (7 (\frac{x^{\frac{1}{3}}}{6} + 5))}{7} - 27000$

Cancel the common factor.

Divide $16200 (\frac{x^{\frac{1}{3}}}{6} + 5)$ by $1$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - 5400 x^{\frac{1}{3}} - 81000 + 16200 (\frac{x^{\frac{1}{3}}}{6} + 5) - 27000$

Apply the distributive property.

Cancel the common factor of $6$ .

Factor $6$ out of $16200$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - 5400 x^{\frac{1}{3}} - 81000 + 6 (2700) (\frac{x^{\frac{1}{3}}}{6}) + 16200 \cdot 5 - 27000$

Cancel the common factor.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - 5400 x^{\frac{1}{3}} - 81000 + 6 \cdot (2700 (\frac{x^{\frac{1}{3}}}{6})) + 16200 \cdot 5 - 27000$

Rewrite the expression.

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - 5400 x^{\frac{1}{3}} - 81000 + 2700 x^{\frac{1}{3}} + 16200 \cdot 5 - 27000$

Multiply $16200$ by $5$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - 5400 x^{\frac{1}{3}} - 81000 + 2700 x^{\frac{1}{3}} + 81000 - 27000$

Simplify by adding terms.

Combine the opposite terms in $x + 90 x^{\frac{2}{3}} + 2700 x^{\frac{1}{3}} + 27000 - 90 x^{\frac{2}{3}} - 5400 x^{\frac{1}{3}} - 81000 + 2700 x^{\frac{1}{3}} + 81000 - 27000$ .

Subtract $90 x^{\frac{2}{3}}$ from $90 x^{\frac{2}{3}}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 2700 x^{\frac{1}{3}} + 27000 + 0 - 5400 x^{\frac{1}{3}} - 81000 + 2700 x^{\frac{1}{3}} + 81000 - 27000$

Add $x + 2700 x^{\frac{1}{3}} + 27000$ and $0$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 2700 x^{\frac{1}{3}} + 27000 - 5400 x^{\frac{1}{3}} - 81000 + 2700 x^{\frac{1}{3}} + 81000 - 27000$

Add $- 81000$ and $81000$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 2700 x^{\frac{1}{3}} + 27000 - 5400 x^{\frac{1}{3}} + 2700 x^{\frac{1}{3}} + 0 - 27000$

Add $x + 2700 x^{\frac{1}{3}} + 27000 - 5400 x^{\frac{1}{3}} + 2700 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 2700 x^{\frac{1}{3}} + 27000 - 5400 x^{\frac{1}{3}} + 2700 x^{\frac{1}{3}} - 27000$

Subtract $27000$ from $27000$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 2700 x^{\frac{1}{3}} - 5400 x^{\frac{1}{3}} + 2700 x^{\frac{1}{3}} + 0$

Add $x + 2700 x^{\frac{1}{3}} - 5400 x^{\frac{1}{3}} + 2700 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 2700 x^{\frac{1}{3}} - 5400 x^{\frac{1}{3}} + 2700 x^{\frac{1}{3}}$

Subtract $5400 x^{\frac{1}{3}}$ from $2700 x^{\frac{1}{3}}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x - 2700 x^{\frac{1}{3}} + 2700 x^{\frac{1}{3}}$

Combine the opposite terms in $x - 2700 x^{\frac{1}{3}} + 2700 x^{\frac{1}{3}}$ .

Add $- 2700 x^{\frac{1}{3}}$ and $2700 x^{\frac{1}{3}}$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (7 (\frac{x^{\frac{1}{3}}}{6} + 5)) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Simplify each term.

Simplify the numerator.

To write $- \frac{3240 x^{2}}{49}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} \cdot \frac{7}{7} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Write each expression with a common denominator of $343$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3240 x^{2}}{49}$ by $\frac{7}{7}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{3}}{343} - \frac{3240 x^{2} \cdot 7}{49 \cdot 7} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Multiply $49$ by $7$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{3}}{343} - \frac{3240 x^{2} \cdot 7}{343} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Combine the numerators over the common denominator.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{3} - 3240 x^{2} \cdot 7}{343} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Simplify the numerator.

Factor $216 x^{2}$ out of $216 x^{3} - 3240 x^{2} \cdot 7$ .

Factor $216 x^{2}$ out of $216 x^{3}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{2} (x) - 3240 x^{2} \cdot 7}{343} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Factor $216 x^{2}$ out of $- 3240 x^{2} \cdot 7$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{2} (x) + 216 x^{2} (- 15 \cdot 7)}{343} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Factor $216 x^{2}$ out of $216 x^{2} (x) + 216 x^{2} (- 15 \cdot 7)$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{2} (x - 15 \cdot 7)}{343} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Multiply $- 15$ by $7$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{2} (x - 105)}{343} + \frac{16200 x}{7} - 27000)}^{\frac{1}{3}}}{6} + 5)$

To write $\frac{16200 x}{7}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{2} (x - 105)}{343} + \frac{16200 x}{7} \cdot \frac{49}{49} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Write each expression with a common denominator of $343$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{16200 x}{7}$ by $\frac{49}{49}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{2} (x - 105)}{343} + \frac{16200 x \cdot 49}{7 \cdot 49} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Multiply $7$ by $49$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{2} (x - 105)}{343} + \frac{16200 x \cdot 49}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Combine the numerators over the common denominator.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x^{2} (x - 105) + 16200 x \cdot 49}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Simplify the numerator.

Factor $216 x$ out of $216 x^{2} (x - 105) + 16200 x \cdot 49$ .

Factor $216 x$ out of $216 x^{2} (x - 105)$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x (x - 105)) + 16200 x \cdot 49}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Factor $216 x$ out of $16200 x \cdot 49$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x (x - 105)) + 216 x (75 \cdot 49)}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Factor $216 x$ out of $216 x (x (x - 105)) + 216 x (75 \cdot 49)$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x (x - 105) + 75 \cdot 49)}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Multiply $75$ by $49$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x (x - 105) + 3675)}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Simplify each term.

Apply the distributive property.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x \cdot x + x \cdot - 105 + 3675)}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Multiply $x$ by $x$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x^{2} + x \cdot - 105 + 3675)}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

Move $- 105$ to the left of $x$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x^{2} - 105 x + 3675)}{343} - 27000)}^{\frac{1}{3}}}{6} + 5)$

To write $- 27000$ as a fraction with a common denominator, multiply by $\frac{343}{343}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x^{2} - 105 x + 3675)}{343} - 27000 \cdot \frac{343}{343})}^{\frac{1}{3}}}{6} + 5)$

Combine $- 27000$ and $\frac{343}{343}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x^{2} - 105 x + 3675)}{343} + \frac{- 27000 \cdot 343}{343})}^{\frac{1}{3}}}{6} + 5)$

Combine the numerators over the common denominator.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 x (x^{2} - 105 x + 3675) - 27000 \cdot 343}{343})}^{\frac{1}{3}}}{6} + 5)$

Simplify the numerator.

Factor $216$ out of $216 x (x^{2} - 105 x + 3675) - 27000 \cdot 343$ .

Factor $216$ out of $216 x (x^{2} - 105 x + 3675)$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x (x^{2} - 105 x + 3675)) - 27000 \cdot 343}{343})}^{\frac{1}{3}}}{6} + 5)$

Factor $216$ out of $- 27000 \cdot 343$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x (x^{2} - 105 x + 3675)) + 216 (- 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Factor $216$ out of $216 (x (x^{2} - 105 x + 3675)) + 216 (- 125 \cdot 343)$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x (x^{2} - 105 x + 3675) - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Apply the distributive property.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x \cdot x^{2} + x (- 105 x) + x \cdot 3675 - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Simplify.

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

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

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x \cdot x^{2} + x (- 105 x) + x \cdot 3675 - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

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

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x^{1 + 2} + x (- 105 x) + x \cdot 3675 - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Add $1$ and $2$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x^{3} + x (- 105 x) + x \cdot 3675 - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Rewrite using the commutative property of multiplication.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x^{3} - 105 x \cdot x + x \cdot 3675 - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Move $3675$ to the left of $x$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x^{3} - 105 x \cdot x + 3675 \cdot x - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x^{3} - 105 (x \cdot x) + 3675 \cdot x - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Multiply $x$ by $x$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x^{3} - 105 x^{2} + 3675 \cdot x - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x^{3} - 105 x^{2} + 3675 x - 125 \cdot 343)}{343})}^{\frac{1}{3}}}{6} + 5)$

Multiply $- 125$ by $343$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 (x^{3} - 105 x^{2} + 3675 x - 42875)}{343})}^{\frac{1}{3}}}{6} + 5)$

Rewrite $x^{3} - 105 x^{2} + 3675 x - 42875$ in a factored form.

Factor $x^{3} - 105 x^{2} + 3675 x - 42875$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 42875, \pm 5, \pm 8575, \pm 7, \pm 6125, \pm 25, \pm 1715, \pm 35, \pm 1225, \pm 49, \pm 875, \pm 125, \pm 343, \pm 175, \pm 245 q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 42875, \pm 5, \pm 8575, \pm 7, \pm 6125, \pm 25, \pm 1715, \pm 35, \pm 1225, \pm 49, \pm 875, \pm 125, \pm 343, \pm 175, \pm 245$

Substitute $35$ and simplify the expression. In this case, the expression is equal to $0$ so $35$ is a root of the polynomial.

Substitute $35$ into the polynomial.

$35^{3} - 105 \cdot 35^{2} + 3675 \cdot 35 - 42875$

Raise $35$ to the power of $3$ .

$42875 - 105 \cdot 35^{2} + 3675 \cdot 35 - 42875$

Raise $35$ to the power of $2$ .

$42875 - 105 \cdot 1225 + 3675 \cdot 35 - 42875$

Multiply $- 105$ by $1225$ .

$42875 - 128625 + 3675 \cdot 35 - 42875$

Subtract $128625$ from $42875$ .

$- 85750 + 3675 \cdot 35 - 42875$

Multiply $3675$ by $35$ .

$- 85750 + 128625 - 42875$

Add $- 85750$ and $128625$ .

$42875 - 42875$

Subtract $42875$ from $42875$ .

$0$

Since $35$ is a known root, divide the polynomial by $x - 35$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 105 x^{2} + 3675 x - 42875}{x - 35}$

Divide $x^{3} - 105 x^{2} + 3675 x - 42875$ by $x - 35$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$35$

$x^{3}$

$105 x^{2}$

$3675 x$

$42875$

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			+	$x^{3}$	-	$35 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 35 x^{2}$

				$x^{2}$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$

Divide the highest order term in the dividend $- 70 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$70 x$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$70 x$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$
					-	$70 x^{2}$	+	$2450 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 70 x^{2} + 2450 x$

				$x^{2}$	-	$70 x$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$
					+	$70 x^{2}$	-	$2450 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$70 x$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$
					+	$70 x^{2}$	-	$2450 x$
							+	$1225 x$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$70 x$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$
					+	$70 x^{2}$	-	$2450 x$
							+	$1225 x$	-	$42875$

Divide the highest order term in the dividend $1225 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$70 x$	+	$1225$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$
					+	$70 x^{2}$	-	$2450 x$
							+	$1225 x$	-	$42875$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$70 x$	+	$1225$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$
					+	$70 x^{2}$	-	$2450 x$
							+	$1225 x$	-	$42875$
							+	$1225 x$	-	$42875$

The expression needs to be subtracted from the dividend, so change all the signs in $1225 x - 42875$

				$x^{2}$	-	$70 x$	+	$1225$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$
					+	$70 x^{2}$	-	$2450 x$
							+	$1225 x$	-	$42875$
							-	$1225 x$	+	$42875$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$70 x$	+	$1225$
$x$	-	$35$		$x^{3}$	-	$105 x^{2}$	+	$3675 x$	-	$42875$
			-	$x^{3}$	+	$35 x^{2}$
					-	$70 x^{2}$	+	$3675 x$
					+	$70 x^{2}$	-	$2450 x$
							+	$1225 x$	-	$42875$
							-	$1225 x$	+	$42875$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 70 x + 1225$

Write $x^{3} - 105 x^{2} + 3675 x - 42875$ as a set of factors.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 ((x - 35) (x^{2} - 70 x + 1225))}{343})}^{\frac{1}{3}}}{6} + 5)$

Factor using the perfect square rule.

Rewrite $1225$ as $35^{2}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 ((x - 35) (x^{2} - 70 x + 35^{2}))}{343})}^{\frac{1}{3}}}{6} + 5)$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$70 x = 2 \cdot x \cdot 35$

Rewrite the polynomial.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 ((x - 35) (x^{2} - 2 \cdot x \cdot 35 + 35^{2}))}{343})}^{\frac{1}{3}}}{6} + 5)$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 35$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 ((x - 35) {(x - 35)}^{2})}{343})}^{\frac{1}{3}}}{6} + 5)$

Combine like factors.

Raise $x - 35$ to the power of $1$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 ((x - 35) {(x - 35)}^{2})}{343})}^{\frac{1}{3}}}{6} + 5)$

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

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 {(x - 35)}^{1 + 2}}{343})}^{\frac{1}{3}}}{6} + 5)$

Add $1$ and $2$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{{(\frac{216 {(x - 35)}^{3}}{343})}^{\frac{1}{3}}}{6} + 5)$

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

Apply the product rule to $\frac{216 {(x - 35)}^{3}}{343}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{{(216 {(x - 35)}^{3})}^{\frac{1}{3}}}{343^{\frac{1}{3}}}}{6} + 5)$

Apply the product rule to $216 {(x - 35)}^{3}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{216^{\frac{1}{3}} {({(x - 35)}^{3})}^{\frac{1}{3}}}{343^{\frac{1}{3}}}}{6} + 5)$

Simplify the numerator.

Rewrite $216$ as $6^{3}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{{(6^{3})}^{\frac{1}{3}} {({(x - 35)}^{3})}^{\frac{1}{3}}}{343^{\frac{1}{3}}}}{6} + 5)$

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

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6^{3 (\frac{1}{3})} {({(x - 35)}^{3})}^{\frac{1}{3}}}{343^{\frac{1}{3}}}}{6} + 5)$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6^{3 (\frac{1}{3})} {({(x - 35)}^{3})}^{\frac{1}{3}}}{343^{\frac{1}{3}}}}{6} + 5)$

Rewrite the expression.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 {({(x - 35)}^{3})}^{\frac{1}{3}}}{343^{\frac{1}{3}}}}{6} + 5)$

Evaluate the exponent.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 {({(x - 35)}^{3})}^{\frac{1}{3}}}{343^{\frac{1}{3}}}}{6} + 5)$

Multiply the exponents in ${({(x - 35)}^{3})}^{\frac{1}{3}}$ .

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

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 {(x - 35)}^{3 (\frac{1}{3})}}{343^{\frac{1}{3}}}}{6} + 5)$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 {(x - 35)}^{3 (\frac{1}{3})}}{343^{\frac{1}{3}}}}{6} + 5)$

Rewrite the expression.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 (x - 35)}{343^{\frac{1}{3}}}}{6} + 5)$

Simplify.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 (x - 35)}{343^{\frac{1}{3}}}}{6} + 5)$

Simplify the denominator.

Rewrite $343$ as $7^{3}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 (x - 35)}{{(7^{3})}^{\frac{1}{3}}}}{6} + 5)$

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

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 (x - 35)}{7^{3 (\frac{1}{3})}}}{6} + 5)$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 (x - 35)}{7^{3 (\frac{1}{3})}}}{6} + 5)$

Rewrite the expression.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 (x - 35)}{7}}{6} + 5)$

Evaluate the exponent.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{\frac{6 (x - 35)}{7}}{6} + 5)$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{6 (x - 35)}{7} \cdot \frac{1}{6} + 5)$

Cancel the common factor of $6$ .

Cancel the common factor.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{6 (x - 35)}{7} \cdot \frac{1}{6} + 5)$

Rewrite the expression.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{x - 35}{7} + 5)$

To write $5$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{x - 35}{7} + 5 \cdot \frac{7}{7})$

Simplify terms.

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

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{x - 35}{7} + \frac{5 \cdot 7}{7})$

Combine the numerators over the common denominator.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{x - 35 + 5 \cdot 7}{7})$

Simplify the numerator.

Multiply $5$ by $7$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{x - 35 + 35}{7})$

Add $- 35$ and $35$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{x + 0}{7})$

Add $x$ and $0$ .

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{x}{7})$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = 7 (\frac{x}{7})$

Rewrite the expression.

$f (\frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000$ is the inverse of $f (x) = 7 (\frac{x^{\frac{1}{3}}}{6} + 5)$ .

$f^{- 1} (x) = \frac{216 x^{3}}{343} - \frac{3240 x^{2}}{49} + \frac{16200 x}{7} - 27000$

Find the Inverse f(x)=7((x^(1/3))/6+5)

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