Find the Inverse f(x)=(6 cube root of x+1)/5

$f (x) = \frac{6 \sqrt[3]{x} + 1}{5}$

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

$y = \frac{6 \sqrt[3]{x} + 1}{5}$

Interchange the variables.

$x = \frac{6 \sqrt[3]{y} + 1}{5}$

Solve for $y$ .

Rewrite the equation as $\frac{6 \sqrt[3]{y} + 1}{5} = x$ .

$\frac{6 \sqrt[3]{y} + 1}{5} = x$

Multiply both sides by $5$ .

$\frac{6 \sqrt[3]{y} + 1}{5} \cdot 5 = x \cdot 5$

Simplify.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{6 \sqrt[3]{y} + 1}{5} \cdot 5 = x \cdot 5$

Rewrite the expression.

$6 \sqrt[3]{y} + 1 = x \cdot 5$

Simplify the right side.

Move $5$ to the left of $x$ .

$6 \sqrt[3]{y} + 1 = 5 x$

Solve for $y$ .

Subtract $1$ from both sides of the equation.

$6 \sqrt[3]{y} = 5 x - 1$

To remove the radical on the left side of the equation, cube both sides of the equation.

${(6 \sqrt[3]{y})}^{3} = {(5 x - 1)}^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

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

Simplify the left side.

Simplify ${(6 y^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $6 y^{\frac{1}{3}}$ .

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

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

$216 {(y^{\frac{1}{3}})}^{3} = {(5 x - 1)}^{3}$

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

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

$216 y^{\frac{1}{3} \cdot 3} = {(5 x - 1)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$216 y^{\frac{1}{3} \cdot 3} = {(5 x - 1)}^{3}$

Rewrite the expression.

$216 y^{1} = {(5 x - 1)}^{3}$

Simplify.

$216 y = {(5 x - 1)}^{3}$

Simplify the right side.

Simplify ${(5 x - 1)}^{3}$ .

Use the Binomial Theorem.

$216 y = {(5 x)}^{3} + 3 {(5 x)}^{2} \cdot - 1 + 3 (5 x) {(- 1)}^{2} + {(- 1)}^{3}$

Simplify each term.

Apply the product rule to $5 x$ .

$216 y = 5^{3} x^{3} + 3 {(5 x)}^{2} \cdot - 1 + 3 (5 x) {(- 1)}^{2} + {(- 1)}^{3}$

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

$216 y = 125 x^{3} + 3 {(5 x)}^{2} \cdot - 1 + 3 (5 x) {(- 1)}^{2} + {(- 1)}^{3}$

Apply the product rule to $5 x$ .

$216 y = 125 x^{3} + 3 (5^{2} x^{2}) \cdot - 1 + 3 (5 x) {(- 1)}^{2} + {(- 1)}^{3}$

Raise $5$ to the power of $2$ .

$216 y = 125 x^{3} + 3 (25 x^{2}) \cdot - 1 + 3 (5 x) {(- 1)}^{2} + {(- 1)}^{3}$

Multiply $25$ by $3$ .

$216 y = 125 x^{3} + 75 x^{2} \cdot - 1 + 3 (5 x) {(- 1)}^{2} + {(- 1)}^{3}$

Multiply $- 1$ by $75$ .

$216 y = 125 x^{3} - 75 x^{2} + 3 (5 x) {(- 1)}^{2} + {(- 1)}^{3}$

Multiply $5$ by $3$ .

$216 y = 125 x^{3} - 75 x^{2} + 15 x {(- 1)}^{2} + {(- 1)}^{3}$

Raise $- 1$ to the power of $2$ .

$216 y = 125 x^{3} - 75 x^{2} + 15 x \cdot 1 + {(- 1)}^{3}$

Multiply $15$ by $1$ .

$216 y = 125 x^{3} - 75 x^{2} + 15 x + {(- 1)}^{3}$

Raise $- 1$ to the power of $3$ .

$216 y = 125 x^{3} - 75 x^{2} + 15 x - 1$

Divide each term in $216 y = 125 x^{3} - 75 x^{2} + 15 x - 1$ by $216$ and simplify.

Divide each term in $216 y = 125 x^{3} - 75 x^{2} + 15 x - 1$ by $216$ .

$\frac{216 y}{216} = \frac{125 x^{3}}{216} + \frac{- 75 x^{2}}{216} + \frac{15 x}{216} + \frac{- 1}{216}$

Simplify the left side.

Cancel the common factor of $216$ .

Cancel the common factor.

$\frac{216 y}{216} = \frac{125 x^{3}}{216} + \frac{- 75 x^{2}}{216} + \frac{15 x}{216} + \frac{- 1}{216}$

Divide $y$ by $1$ .

$y = \frac{125 x^{3}}{216} + \frac{- 75 x^{2}}{216} + \frac{15 x}{216} + \frac{- 1}{216}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 75$ and $216$ .

Factor $3$ out of $- 75 x^{2}$ .

$y = \frac{125 x^{3}}{216} + \frac{3 (- 25 x^{2})}{216} + \frac{15 x}{216} + \frac{- 1}{216}$

Cancel the common factors.

Factor $3$ out of $216$ .

$y = \frac{125 x^{3}}{216} + \frac{3 (- 25 x^{2})}{3 (72)} + \frac{15 x}{216} + \frac{- 1}{216}$

Cancel the common factor.

$y = \frac{125 x^{3}}{216} + \frac{3 (- 25 x^{2})}{3 \cdot 72} + \frac{15 x}{216} + \frac{- 1}{216}$

Rewrite the expression.

$y = \frac{125 x^{3}}{216} + \frac{- 25 x^{2}}{72} + \frac{15 x}{216} + \frac{- 1}{216}$

Move the negative in front of the fraction.

$y = \frac{125 x^{3}}{216} - \frac{(25) x^{2}}{72} + \frac{15 x}{216} + \frac{- 1}{216}$

Cancel the common factor of $15$ and $216$ .

Factor $3$ out of $15 x$ .

$y = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{3 (5 x)}{216} + \frac{- 1}{216}$

Cancel the common factors.

Factor $3$ out of $216$ .

$y = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{3 (5 x)}{3 (72)} + \frac{- 1}{216}$

Cancel the common factor.

$y = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{3 (5 x)}{3 \cdot 72} + \frac{- 1}{216}$

Rewrite the expression.

$y = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} + \frac{- 1}{216}$

Move the negative in front of the fraction.

$y = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}$

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

$f^{- 1} (x) = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}$

Verify if $f^{- 1} (x) = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}$ is the inverse of $f (x) = \frac{6 \sqrt[3]{x} + 1}{5}$ .

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

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} - \frac{25 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{2}}{72} + \frac{5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72} - \frac{1}{216}$

Simplify terms.

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 25 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{2} + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Simplify each term.

Apply the product rule to $\frac{6 \sqrt[3]{x} + 1}{5}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 25 \frac{{(6 \sqrt[3]{x} + 1)}^{2}}{5^{2}} + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Raise $5$ to the power of $2$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 25 \frac{{(6 \sqrt[3]{x} + 1)}^{2}}{25} + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Cancel the common factor of $25$ .

Factor $25$ out of $- 25$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{25 (- 1) (\frac{{(6 \sqrt[3]{x} + 1)}^{2}}{25}) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Cancel the common factor.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{25 \cdot (- 1 \frac{{(6 \sqrt[3]{x} + 1)}^{2}}{25}) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Rewrite the expression.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 {(6 \sqrt[3]{x} + 1)}^{2} + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Rewrite ${(6 \sqrt[3]{x} + 1)}^{2}$ as $(6 \sqrt[3]{x} + 1) (6 \sqrt[3]{x} + 1)$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 ((6 \sqrt[3]{x} + 1) (6 \sqrt[3]{x} + 1)) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Expand $(6 \sqrt[3]{x} + 1) (6 \sqrt[3]{x} + 1)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (6 \sqrt[3]{x} (6 \sqrt[3]{x} + 1) + 1 (6 \sqrt[3]{x} + 1)) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Apply the distributive property.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (6 \sqrt[3]{x} (6 \sqrt[3]{x}) + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x} + 1)) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply $6 \sqrt[3]{x} (6 \sqrt[3]{x})$ .

Multiply $6$ by $6$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 \sqrt[3]{x} \sqrt[3]{x} + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Raise $\sqrt[3]{x}$ to the power of $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 (\sqrt[3]{x} \sqrt[3]{x}) + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Raise $\sqrt[3]{x}$ to the power of $1$ .

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

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 {\sqrt[3]{x}}^{1 + 1} + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Add $1$ and $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 {\sqrt[3]{x}}^{2} + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Rewrite ${\sqrt[3]{x}}^{2}$ as ${(x^{2})}^{\frac{1}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 \sqrt[3]{x^{2}} + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Multiply $6$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 \sqrt[3]{x^{2}} + 6 \sqrt[3]{x} + 1 (6 \sqrt[3]{x}) + 1 \cdot 1) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Multiply $6 \sqrt[3]{x}$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 \sqrt[3]{x^{2}} + 6 \sqrt[3]{x} + 6 \sqrt[3]{x} + 1 \cdot 1) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Multiply $1$ by $1$ .

Add $6 \sqrt[3]{x}$ and $6 \sqrt[3]{x}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 \sqrt[3]{x^{2}} + 12 \sqrt[3]{x} + 1) + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Apply the distributive property.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 1 (36 \sqrt[3]{x^{2}}) - 1 (12 \sqrt[3]{x}) - 1 \cdot 1 + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Simplify.

Multiply $36$ by $- 1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 1 (12 \sqrt[3]{x}) - 1 \cdot 1 + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Multiply $12$ by $- 1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 12 \sqrt[3]{x} - 1 \cdot 1 + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Multiply $- 1$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 12 \sqrt[3]{x} - 1 + 5 (\frac{6 \sqrt[3]{x} + 1}{5})}{72}$

Cancel the common factor of $5$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 12 \sqrt[3]{x} - 1 + 6 \sqrt[3]{x} + 1}{72}$

Simplify by adding terms.

Combine the opposite terms in $- 36 \sqrt[3]{x^{2}} - 12 \sqrt[3]{x} - 1 + 6 \sqrt[3]{x} + 1$ .

Add $- 1$ and $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 12 \sqrt[3]{x} + 0 + 6 \sqrt[3]{x}}{72}$

Add $- 36 \sqrt[3]{x^{2}} - 12 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 12 \sqrt[3]{x} + 6 \sqrt[3]{x}}{72}$

Add $- 12 \sqrt[3]{x}$ and $6 \sqrt[3]{x}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 {(\frac{6 \sqrt[3]{x} + 1}{5})}^{3}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}}{72}$

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{6 \sqrt[3]{x} + 1}{5}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 (\frac{{(6 \sqrt[3]{x} + 1)}^{3}}{5^{3}})}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}}{72}$

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

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{125 (\frac{{(6 \sqrt[3]{x} + 1)}^{3}}{125})}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}}{72}$

Combine $125$ and $\frac{{(6 \sqrt[3]{x} + 1)}^{3}}{125}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{\frac{125 {(6 \sqrt[3]{x} + 1)}^{3}}{125}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}}{72}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{125 {(6 \sqrt[3]{x} + 1)}^{3}}{125}$ by cancelling the common factors.

Cancel the common factor.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{\frac{125 {(6 \sqrt[3]{x} + 1)}^{3}}{125}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}}{72}$

Rewrite the expression.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{\frac{{(6 \sqrt[3]{x} + 1)}^{3}}{1}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}}{72}$

Divide ${(6 \sqrt[3]{x} + 1)}^{3}$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} + \frac{- 1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}}{72}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}}{72}$

Cancel the common factor of $- 36 \sqrt[3]{x^{2}} - 6 \sqrt[3]{x}$ and $72$ .

Factor $6$ out of $- 36 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{6 (- 6 \sqrt[3]{x^{2}}) - 6 \sqrt[3]{x}}{72}$

Factor $6$ out of $- 6 \sqrt[3]{x}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{6 (- 6 \sqrt[3]{x^{2}}) + 6 (- \sqrt[3]{x})}{72}$

Factor $6$ out of $6 (- 6 \sqrt[3]{x^{2}}) + 6 (- \sqrt[3]{x})$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{6 (- 6 \sqrt[3]{x^{2}} - \sqrt[3]{x})}{72}$

Cancel the common factors.

Factor $6$ out of $72$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{6 (- 6 \sqrt[3]{x^{2}} - \sqrt[3]{x})}{6 (12)}$

Cancel the common factor.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{6 (- 6 \sqrt[3]{x^{2}} - \sqrt[3]{x})}{6 \cdot 12}$

Rewrite the expression.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- 6 \sqrt[3]{x^{2}} - \sqrt[3]{x}}{12}$

Simplify the numerator.

Factor $- 1$ out of $- 6 \sqrt[3]{x^{2}} - \sqrt[3]{x}$ .

Factor $- 1$ out of $- 6 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- (6 \sqrt[3]{x^{2}}) - \sqrt[3]{x}}{12}$

Factor $- 1$ out of $- \sqrt[3]{x}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- (6 \sqrt[3]{x^{2}}) - (\sqrt[3]{x})}{12}$

Factor $- 1$ out of $- (6 \sqrt[3]{x^{2}}) - (\sqrt[3]{x})$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- (6 \sqrt[3]{x^{2}} + \sqrt[3]{x})}{12}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- (6 x^{\frac{2}{3}} + \sqrt[3]{x})}{12}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- (6 x^{\frac{2}{3}} + x^{\frac{1}{3}})}{12}$

Factor $x^{\frac{1}{3}}$ out of $6 x^{\frac{2}{3}} + x^{\frac{1}{3}}$ .

Factor $x^{\frac{1}{3}}$ out of $6 x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- (x^{\frac{1}{3}} (6 x^{\frac{1}{3}}) + x^{\frac{1}{3}})}{12}$

Multiply by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- (x^{\frac{1}{3}} (6 x^{\frac{1}{3}}) + x^{\frac{1}{3}} \cdot 1)}{12}$

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} (6 x^{\frac{1}{3}}) + x^{\frac{1}{3}} \cdot 1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- (x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1))}{12}$

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} + \frac{- x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3}}{216} - \frac{1}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3} - 1}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Simplify each term.

Simplify the numerator.

Rewrite $1$ as $1^{3}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{{(6 \sqrt[3]{x} + 1)}^{3} - 1^{3}}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 6 \sqrt[3]{x} + 1$ and $b = 1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{(6 \sqrt[3]{x} + 1 - 1) ({(6 \sqrt[3]{x} + 1)}^{2} + (6 \sqrt[3]{x} + 1) \cdot 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Simplify.

Subtract $1$ from $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{(6 \sqrt[3]{x} + 0) ({(6 \sqrt[3]{x} + 1)}^{2} + (6 \sqrt[3]{x} + 1) \cdot 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Add $6 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} ({(6 \sqrt[3]{x} + 1)}^{2} + (6 \sqrt[3]{x} + 1) \cdot 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Multiply $6 \sqrt[3]{x} + 1$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} ({(6 \sqrt[3]{x} + 1)}^{2} + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Simplify each term.

Rewrite ${(6 \sqrt[3]{x} + 1)}^{2}$ as $(6 \sqrt[3]{x} + 1) (6 \sqrt[3]{x} + 1)$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} ((6 \sqrt[3]{x} + 1) (6 \sqrt[3]{x} + 1) + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Expand $(6 \sqrt[3]{x} + 1) (6 \sqrt[3]{x} + 1)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (6 \sqrt[3]{x} (6 \sqrt[3]{x} + 1) + 1 (6 \sqrt[3]{x} + 1) + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Apply the distributive property.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (6 \sqrt[3]{x} (6 \sqrt[3]{x}) + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x} + 1) + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Apply the distributive property.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (6 \sqrt[3]{x} (6 \sqrt[3]{x}) + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Simplify and combine like terms.

Simplify each term.

Multiply $6 \sqrt[3]{x} (6 \sqrt[3]{x})$ .

Multiply $6$ by $6$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x} \sqrt[3]{x} + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Raise $\sqrt[3]{x}$ to the power of $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 (\sqrt[3]{x} \sqrt[3]{x}) + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Raise $\sqrt[3]{x}$ to the power of $1$ .

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

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 {\sqrt[3]{x}}^{1 + 1} + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Add $1$ and $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 {\sqrt[3]{x}}^{2} + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Rewrite ${\sqrt[3]{x}}^{2}$ as ${(x^{2})}^{\frac{1}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 6 \sqrt[3]{x} \cdot 1 + 1 (6 \sqrt[3]{x}) + 1 \cdot 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Multiply $6$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 6 \sqrt[3]{x} + 1 (6 \sqrt[3]{x}) + 1 \cdot 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Multiply $6 \sqrt[3]{x}$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 6 \sqrt[3]{x} + 6 \sqrt[3]{x} + 1 \cdot 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Multiply $1$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 6 \sqrt[3]{x} + 6 \sqrt[3]{x} + 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Add $6 \sqrt[3]{x}$ and $6 \sqrt[3]{x}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 12 \sqrt[3]{x} + 1 + 6 \sqrt[3]{x} + 1 + 1^{2})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

One to any power is one.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 12 \sqrt[3]{x} + 1 + 6 \sqrt[3]{x} + 1 + 1)}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Add $12 \sqrt[3]{x}$ and $6 \sqrt[3]{x}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 1 + 18 \sqrt[3]{x} + 1 + 1)}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Add $1$ and $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 2 + 18 \sqrt[3]{x} + 1)}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Add $2$ and $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (36 \sqrt[3]{x^{2}} + 3 + 18 \sqrt[3]{x})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Factor $3$ out of $36 \sqrt[3]{x^{2}} + 3 + 18 \sqrt[3]{x}$ .

Factor $3$ out of $36 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (3 (12 \sqrt[3]{x^{2}}) + 3 + 18 \sqrt[3]{x})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Factor $3$ out of $3$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (3 (12 \sqrt[3]{x^{2}}) + 3 (1) + 18 \sqrt[3]{x})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Factor $3$ out of $18 \sqrt[3]{x}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (3 (12 \sqrt[3]{x^{2}}) + 3 (1) + 3 (6 \sqrt[3]{x}))}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Factor $3$ out of $3 (12 \sqrt[3]{x^{2}}) + 3 (1)$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (3 (12 \sqrt[3]{x^{2}} + 1) + 3 (6 \sqrt[3]{x}))}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Factor $3$ out of $3 (12 \sqrt[3]{x^{2}} + 1) + 3 (6 \sqrt[3]{x})$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} (3 (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x}))}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{6 \sqrt[3]{x} \cdot (3 (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x}))}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Multiply $3$ by $6$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{18 \sqrt[3]{x} (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x})}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Cancel the common factor of $18$ and $216$ .

Factor $18$ out of $18 \sqrt[3]{x} (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x})$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{18 (\sqrt[3]{x} (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x}))}{216} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Cancel the common factors.

Factor $18$ out of $216$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{18 (\sqrt[3]{x} (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x}))}{18 \cdot 12} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Cancel the common factor.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{18 (\sqrt[3]{x} (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x}))}{18 \cdot 12} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Rewrite the expression.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{\sqrt[3]{x} (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x})}{12} - \frac{x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{\sqrt[3]{x} (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x}) - x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 \sqrt[3]{x^{2}} + 1 + 6 \sqrt[3]{x}) - x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 \sqrt[3]{x}) - x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 x^{\frac{1}{3}}) - x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)}{12}$

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 x^{\frac{1}{3}}) - x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)$ .

Factor $x^{\frac{1}{3}}$ out of $- x^{\frac{1}{3}} (6 x^{\frac{1}{3}} + 1)$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 x^{\frac{1}{3}}) + x^{\frac{1}{3}} (- (6 x^{\frac{1}{3}} + 1))}{12}$

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 x^{\frac{1}{3}}) + x^{\frac{1}{3}} (- (6 x^{\frac{1}{3}} + 1))$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 x^{\frac{1}{3}} - (6 x^{\frac{1}{3}} + 1))}{12}$

Apply the distributive property.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 x^{\frac{1}{3}} - (6 x^{\frac{1}{3}}) - 1 \cdot 1)}{12}$

Multiply $6$ by $- 1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 x^{\frac{1}{3}} - 6 x^{\frac{1}{3}} - 1 \cdot 1)}{12}$

Multiply $- 1$ by $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 1 + 6 x^{\frac{1}{3}} - 6 x^{\frac{1}{3}} - 1)}{12}$

Subtract $1$ from $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 6 x^{\frac{1}{3}} - 6 x^{\frac{1}{3}} + 0)}{12}$

Add $12 x^{\frac{2}{3}} + 6 x^{\frac{1}{3}} - 6 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 6 x^{\frac{1}{3}} - 6 x^{\frac{1}{3}})}{12}$

Subtract $6 x^{\frac{1}{3}}$ from $6 x^{\frac{1}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} (12 x^{\frac{2}{3}} + 0)}{12}$

Add $12 x^{\frac{2}{3}}$ and $0$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{1}{3}} \cdot (12 x^{\frac{2}{3}})}{12}$

Combine exponents.

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

Move $x^{\frac{2}{3}}$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{2}{3}} x^{\frac{1}{3}} \cdot 12}{12}$

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

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{2}{3} + \frac{1}{3}} \cdot 12}{12}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{2 + 1}{3}} \cdot 12}{12}$

Add $2$ and $1$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x^{\frac{3}{3}} \cdot 12}{12}$

Divide $3$ by $3$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x \cdot 12}{12}$

Simplify $x^{1} \cdot 12$ .

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x \cdot 12}{12}$

Cancel the common factor of $12$ .

Cancel the common factor.

$f^{- 1} (\frac{6 \sqrt[3]{x} + 1}{5}) = \frac{x \cdot 12}{12}$

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

Evaluate $f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 \sqrt[3]{\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}} + 1}{5}$

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}}$ as ${(\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Simplify each term.

To write $- \frac{25 x^{2}}{72}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} \cdot \frac{3}{3} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Write each expression with a common denominator of $216$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{25 x^{2}}{72}$ by $\frac{3}{3}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3}}{216} - \frac{25 x^{2} \cdot 3}{72 \cdot 3} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $72$ by $3$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3}}{216} - \frac{25 x^{2} \cdot 3}{216} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Combine the numerators over the common denominator.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3} - 25 x^{2} \cdot 3}{216} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Simplify the numerator.

Factor $25 x^{2}$ out of $125 x^{3} - 25 x^{2} \cdot 3$ .

Factor $25 x^{2}$ out of $125 x^{3}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{25 x^{2} (5 x) - 25 x^{2} \cdot 3}{216} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Factor $25 x^{2}$ out of $- 25 x^{2} \cdot 3$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{25 x^{2} (5 x) + 25 x^{2} (- 1 \cdot 3)}{216} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Factor $25 x^{2}$ out of $25 x^{2} (5 x) + 25 x^{2} (- 1 \cdot 3)$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{25 x^{2} (5 x - 1 \cdot 3)}{216} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $- 1$ by $3$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{25 x^{2} (5 x - 3)}{216} + \frac{5 x}{72} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

To write $\frac{5 x}{72}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{25 x^{2} (5 x - 3)}{216} + \frac{5 x}{72} \cdot \frac{3}{3} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Write each expression with a common denominator of $216$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 x}{72}$ by $\frac{3}{3}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{25 x^{2} (5 x - 3)}{216} + \frac{5 x \cdot 3}{72 \cdot 3} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $72$ by $3$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{25 x^{2} (5 x - 3)}{216} + \frac{5 x \cdot 3}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Combine the numerators over the common denominator.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{25 x^{2} (5 x - 3) + 5 x \cdot 3}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Simplify the numerator.

Factor $5 x$ out of $25 x^{2} (5 x - 3) + 5 x \cdot 3$ .

Factor $5 x$ out of $25 x^{2} (5 x - 3)$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (5 x (5 x - 3)) + 5 x \cdot 3}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Factor $5 x$ out of $5 x \cdot 3$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (5 x (5 x - 3)) + 5 x (3)}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Factor $5 x$ out of $5 x (5 x (5 x - 3)) + 5 x (3)$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (5 x (5 x - 3) + 3)}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Apply the distributive property.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (5 x (5 x) + 5 x \cdot - 3 + 3)}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Rewrite using the commutative property of multiplication.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (5 \cdot (5 x \cdot x) + 5 x \cdot - 3 + 3)}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $- 3$ by $5$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (5 \cdot (5 x \cdot x) - 15 x + 3)}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Simplify each term.

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

Move $x$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (5 \cdot (5 (x \cdot x)) - 15 x + 3)}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $x$ by $x$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (5 \cdot (5 x^{2}) - 15 x + 3)}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $5$ by $5$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (25 x^{2} - 15 x + 3)}{216} - \frac{1}{216})}^{\frac{1}{3}} + 1}{5}$

Combine the numerators over the common denominator.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (25 x^{2} - 15 x + 3) - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 x (25 x^{2}) + 5 x (- 15 x) + 5 x \cdot 3 - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Simplify.

Rewrite using the commutative property of multiplication.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 \cdot (25 x \cdot x^{2}) + 5 x (- 15 x) + 5 x \cdot 3 - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Rewrite using the commutative property of multiplication.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 \cdot (25 x \cdot x^{2}) + 5 \cdot (- 15 x \cdot x) + 5 x \cdot 3 - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $3$ by $5$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 \cdot (25 x \cdot x^{2}) + 5 \cdot (- 15 x \cdot x) + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Simplify each term.

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

Move $x^{2}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 \cdot (25 (x^{2} x)) + 5 \cdot (- 15 x \cdot x) + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

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

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

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 \cdot (25 (x^{2} x)) + 5 \cdot (- 15 x \cdot x) + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

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

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 \cdot (25 x^{2 + 1}) + 5 \cdot (- 15 x \cdot x) + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Add $2$ and $1$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{5 \cdot (25 x^{3}) + 5 \cdot (- 15 x \cdot x) + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $5$ by $25$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3} + 5 \cdot (- 15 x \cdot x) + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

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

Move $x$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3} + 5 \cdot (- 15 (x \cdot x)) + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $x$ by $x$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3} + 5 \cdot (- 15 x^{2}) + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Multiply $5$ by $- 15$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{125 x^{3} - 75 x^{2} + 15 x - 1}{216})}^{\frac{1}{3}} + 1}{5}$

Rewrite $125 x^{3} - 75 x^{2} + 15 x - 1$ in a factored form.

Factor $125 x^{3} - 75 x^{2} + 15 x - 1$ 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.

$p = \pm 1 q = \pm 1, \pm 125, \pm 5, \pm 25$

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

$\pm 1, \pm 0.008, \pm 0.2, \pm 0.04$

Substitute $0.2$ and simplify the expression. In this case, the expression is equal to $0$ so $0.2$ is a root of the polynomial.

Substitute $0.2$ into the polynomial.

$125 \cdot {0.2}^{3} - 75 \cdot {0.2}^{2} + 15 \cdot 0.2 - 1$

Raise $0.2$ to the power of $3$ .

$125 \cdot 0.008 - 75 \cdot {0.2}^{2} + 15 \cdot 0.2 - 1$

Multiply $125$ by $0.008$ .

$1 - 75 \cdot {0.2}^{2} + 15 \cdot 0.2 - 1$

Raise $0.2$ to the power of $2$ .

$1 - 75 \cdot 0.04 + 15 \cdot 0.2 - 1$

Multiply $- 75$ by $0.04$ .

$1 - 3 + 15 \cdot 0.2 - 1$

Subtract $3$ from $1$ .

$- 2 + 15 \cdot 0.2 - 1$

Multiply $15$ by $0.2$ .

$- 2 + 3 - 1$

Add $- 2$ and $3$ .

$1 - 1$

Subtract $1$ from $1$ .

$0$

Since $0.2$ is a known root, divide the polynomial by $5 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{125 x^{3} - 75 x^{2} + 15 x - 1}{5 x - 1}$

Divide $125 x^{3} - 75 x^{2} + 15 x - 1$ by $5 x - 1$ .

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

$5 x$

$1$

$125 x^{3}$

$75 x^{2}$

$15 x$

$1$

Divide the highest order term in the dividend $125 x^{3}$ by the highest order term in divisor $5 x$ .

					$25 x^{2}$
	$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$

Multiply the new quotient term by the divisor.

				$25 x^{2}$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			+	$125 x^{3}$	-	$25 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $125 x^{3} - 25 x^{2}$

				$25 x^{2}$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$

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

				$25 x^{2}$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$

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

				$25 x^{2}$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$

Divide the highest order term in the dividend $- 50 x^{2}$ by the highest order term in divisor $5 x$ .

				$25 x^{2}$	-	$10 x$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$

Multiply the new quotient term by the divisor.

				$25 x^{2}$	-	$10 x$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$
					-	$50 x^{2}$	+	$10 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 50 x^{2} + 10 x$

				$25 x^{2}$	-	$10 x$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$
					+	$50 x^{2}$	-	$10 x$

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

				$25 x^{2}$	-	$10 x$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$
					+	$50 x^{2}$	-	$10 x$
							+	$5 x$

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

				$25 x^{2}$	-	$10 x$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$
					+	$50 x^{2}$	-	$10 x$
							+	$5 x$	-	$1$

Divide the highest order term in the dividend $5 x$ by the highest order term in divisor $5 x$ .

				$25 x^{2}$	-	$10 x$	+	$1$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$
					+	$50 x^{2}$	-	$10 x$
							+	$5 x$	-	$1$

Multiply the new quotient term by the divisor.

				$25 x^{2}$	-	$10 x$	+	$1$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$
					+	$50 x^{2}$	-	$10 x$
							+	$5 x$	-	$1$
							+	$5 x$	-	$1$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x - 1$

				$25 x^{2}$	-	$10 x$	+	$1$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$
					+	$50 x^{2}$	-	$10 x$
							+	$5 x$	-	$1$
							-	$5 x$	+	$1$

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

				$25 x^{2}$	-	$10 x$	+	$1$
$5 x$	-	$1$		$125 x^{3}$	-	$75 x^{2}$	+	$15 x$	-	$1$
			-	$125 x^{3}$	+	$25 x^{2}$
					-	$50 x^{2}$	+	$15 x$
					+	$50 x^{2}$	-	$10 x$
							+	$5 x$	-	$1$
							-	$5 x$	+	$1$
										$0$

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

$25 x^{2} - 10 x + 1$

Write $125 x^{3} - 75 x^{2} + 15 x - 1$ as a set of factors.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{(5 x - 1) (25 x^{2} - 10 x + 1)}{216})}^{\frac{1}{3}} + 1}{5}$

Factor using the perfect square rule.

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{(5 x - 1) ({(5 x)}^{2} - 10 x + 1)}{216})}^{\frac{1}{3}} + 1}{5}$

Rewrite $1$ as $1^{2}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{(5 x - 1) ({(5 x)}^{2} - 10 x + 1^{2})}{216})}^{\frac{1}{3}} + 1}{5}$

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

$10 x = 2 \cdot (5 x) \cdot 1$

Rewrite the polynomial.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{(5 x - 1) ({(5 x)}^{2} - 2 \cdot (5 x) \cdot 1 + 1^{2})}{216})}^{\frac{1}{3}} + 1}{5}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 5 x$ and $b = 1$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{(5 x - 1) {(5 x - 1)}^{2}}{216})}^{\frac{1}{3}} + 1}{5}$

Combine like factors.

Raise $5 x - 1$ to the power of $1$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{(5 x - 1) {(5 x - 1)}^{2}}{216})}^{\frac{1}{3}} + 1}{5}$

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

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{{(5 x - 1)}^{1 + 2}}{216})}^{\frac{1}{3}} + 1}{5}$

Add $1$ and $2$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 {(\frac{{(5 x - 1)}^{3}}{216})}^{\frac{1}{3}} + 1}{5}$

Apply the product rule to $\frac{{(5 x - 1)}^{3}}{216}$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{{({(5 x - 1)}^{3})}^{\frac{1}{3}}}{216^{\frac{1}{3}}}) + 1}{5}$

Simplify the numerator.

Multiply the exponents in ${({(5 x - 1)}^{3})}^{\frac{1}{3}}$ .

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

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{{(5 x - 1)}^{3 (\frac{1}{3})}}{216^{\frac{1}{3}}}) + 1}{5}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{{(5 x - 1)}^{3 (\frac{1}{3})}}{216^{\frac{1}{3}}}) + 1}{5}$

Rewrite the expression.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{5 x - 1}{216^{\frac{1}{3}}}) + 1}{5}$

Simplify.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{5 x - 1}{216^{\frac{1}{3}}}) + 1}{5}$

Simplify the denominator.

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

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{5 x - 1}{{(6^{3})}^{\frac{1}{3}}}) + 1}{5}$

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

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{5 x - 1}{6^{3 (\frac{1}{3})}}) + 1}{5}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{5 x - 1}{6^{3 (\frac{1}{3})}}) + 1}{5}$

Rewrite the expression.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{5 x - 1}{6}) + 1}{5}$

Evaluate the exponent.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{5 x - 1}{6}) + 1}{5}$

Cancel the common factor of $6$ .

Cancel the common factor.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{6 (\frac{5 x - 1}{6}) + 1}{5}$

Rewrite the expression.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{5 x - 1 + 1}{5}$

Combine the opposite terms in $5 x - 1 + 1$ .

Add $- 1$ and $1$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{5 x + 0}{5}$

Add $5 x$ and $0$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{5 x}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = \frac{5 x}{5}$

Divide $x$ by $1$ .

$f (\frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}$ is the inverse of $f (x) = \frac{6 \sqrt[3]{x} + 1}{5}$ .

$f^{- 1} (x) = \frac{125 x^{3}}{216} - \frac{25 x^{2}}{72} + \frac{5 x}{72} - \frac{1}{216}$

Find the Inverse f(x)=(6 cube root of x+1)/5

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