Find the Inverse 4( cube root of x+10)

$4 (\sqrt[3]{x} + 10)$

Step 1

Interchange the variables.

$x = 4 (\sqrt[3]{y} + 10)$

Step 2

Solve for $y$ .

Rewrite the equation as $4 (\sqrt[3]{y} + 10) = x$ .

$4 (\sqrt[3]{y} + 10) = x$

Divide each term in $4 (\sqrt[3]{y} + 10) = x$ by $4$ and simplify.

Divide each term in $4 (\sqrt[3]{y} + 10) = x$ by $4$ .

$\frac{4 (\sqrt[3]{y} + 10)}{4} = \frac{x}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (\sqrt[3]{y} + 10)}{4} = \frac{x}{4}$

Divide $\sqrt[3]{y} + 10$ by $1$ .

$\sqrt[3]{y} + 10 = \frac{x}{4}$

Subtract $10$ from both sides of the equation.

$\sqrt[3]{y} = \frac{x}{4} - 10$

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

${\sqrt[3]{y}}^{3} = {(\frac{x}{4} - 10)}^{3}$

Simplify each side of the equation.

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

${(y^{\frac{1}{3}})}^{3} = {(\frac{x}{4} - 10)}^{3}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{4} - 10)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{4} - 10)}^{3}$

Rewrite the expression.

$y^{1} = {(\frac{x}{4} - 10)}^{3}$

Simplify.

$y = {(\frac{x}{4} - 10)}^{3}$

Simplify the right side.

Simplify ${(\frac{x}{4} - 10)}^{3}$ .

Use the Binomial Theorem.

$y = {(\frac{x}{4})}^{3} + 3 {(\frac{x}{4})}^{2} \cdot - 10 + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Simplify each term.

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

$y = \frac{x^{3}}{4^{3}} + 3 {(\frac{x}{4})}^{2} \cdot - 10 + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Raise $4$ to the power of $3$ .

$y = \frac{x^{3}}{64} + 3 {(\frac{x}{4})}^{2} \cdot - 10 + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

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

$y = \frac{x^{3}}{64} + 3 \frac{x^{2}}{4^{2}} \cdot - 10 + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Raise $4$ to the power of $2$ .

$y = \frac{x^{3}}{64} + 3 \frac{x^{2}}{16} \cdot - 10 + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Combine $3$ and $\frac{x^{2}}{16}$ .

$y = \frac{x^{3}}{64} + \frac{3 x^{2}}{16} \cdot - 10 + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$y = \frac{x^{3}}{64} + \frac{3 x^{2}}{2 (8)} \cdot - 10 + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Factor $2$ out of $- 10$ .

$y = \frac{x^{3}}{64} + \frac{3 x^{2}}{2 \cdot 8} \cdot (2 \cdot - 5) + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Cancel the common factor.

$y = \frac{x^{3}}{64} + \frac{3 x^{2}}{2 \cdot 8} \cdot (2 \cdot - 5) + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Rewrite the expression.

$y = \frac{x^{3}}{64} + \frac{3 x^{2}}{8} \cdot - 5 + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Combine $\frac{3 x^{2}}{8}$ and $- 5$ .

$y = \frac{x^{3}}{64} + \frac{3 x^{2} \cdot - 5}{8} + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Multiply $- 5$ by $3$ .

$y = \frac{x^{3}}{64} + \frac{- 15 x^{2}}{8} + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Move the negative in front of the fraction.

$y = \frac{x^{3}}{64} - \frac{(15) x^{2}}{8} + 3 \frac{x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Combine $3$ and $\frac{x}{4}$ .

$y = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + \frac{3 x}{4} {(- 10)}^{2} + {(- 10)}^{3}$

Raise $- 10$ to the power of $2$ .

$y = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + \frac{3 x}{4} \cdot 100 + {(- 10)}^{3}$

Cancel the common factor of $4$ .

Factor $4$ out of $100$ .

$y = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + \frac{3 x}{4} \cdot (4 (25)) + {(- 10)}^{3}$

Cancel the common factor.

$y = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + \frac{3 x}{4} \cdot (4 \cdot 25) + {(- 10)}^{3}$

Rewrite the expression.

$y = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 3 x \cdot 25 + {(- 10)}^{3}$

Multiply $25$ by $3$ .

$y = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x + {(- 10)}^{3}$

Raise $- 10$ to the power of $3$ .

$y = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000$

Step 3

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

$f^{- 1} (x) = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000$ is the inverse of $f (x) = 4 (\sqrt[3]{x} + 10)$ .

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 $f^{- 1} (4 (\sqrt[3]{x} + 10))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = \frac{{(4 (\sqrt[3]{x} + 10))}^{3}}{64} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Simplify each term.

Simplify the numerator.

Apply the product rule to $4 (\sqrt[3]{x} + 10)$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = \frac{4^{3} {(\sqrt[3]{x} + 10)}^{3}}{64} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Raise $4$ to the power of $3$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = \frac{64 {(\sqrt[3]{x} + 10)}^{3}}{64} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Cancel the common factor of $64$ .

Cancel the common factor.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = \frac{64 {(\sqrt[3]{x} + 10)}^{3}}{64} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Divide ${(\sqrt[3]{x} + 10)}^{3}$ by $1$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = {(\sqrt[3]{x} + 10)}^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Use the Binomial Theorem.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = {\sqrt[3]{x}}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot 10 + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Simplify each term.

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = {(x^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot 10 + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

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

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x}}^{2} \cdot 10 + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

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

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot 10 + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot 10 + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Rewrite the expression.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 3 {\sqrt[3]{x}}^{2} \cdot 10 + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Simplify.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 3 {\sqrt[3]{x}}^{2} \cdot 10 + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

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

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 3 \sqrt[3]{x^{2}} \cdot 10 + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Multiply $10$ by $3$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 10^{2} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Raise $10$ to the power of $2$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 100 + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Multiply $100$ by $3$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 10^{3} - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Raise $10$ to the power of $3$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - \frac{15 {(4 (\sqrt[3]{x} + 10))}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Simplify the numerator.

Apply the product rule to $4 (\sqrt[3]{x} + 10)$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - \frac{15 \cdot (4^{2} {(\sqrt[3]{x} + 10)}^{2})}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Raise $4$ to the power of $2$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - \frac{15 \cdot (16 {(\sqrt[3]{x} + 10)}^{2})}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Multiply $15$ by $16$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - \frac{240 {(\sqrt[3]{x} + 10)}^{2}}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Cancel the common factor of $240$ and $8$ .

Factor $8$ out of $240 {(\sqrt[3]{x} + 10)}^{2}$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - \frac{8 (30 {(\sqrt[3]{x} + 10)}^{2})}{8} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Cancel the common factors.

Factor $8$ out of $8$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - \frac{8 (30 {(\sqrt[3]{x} + 10)}^{2})}{8 (1)} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Cancel the common factor.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - \frac{8 (30 {(\sqrt[3]{x} + 10)}^{2})}{8 \cdot 1} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Rewrite the expression.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - \frac{30 {(\sqrt[3]{x} + 10)}^{2}}{1} + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Divide $30 {(\sqrt[3]{x} + 10)}^{2}$ by $1$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 {(\sqrt[3]{x} + 10)}^{2}) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Rewrite ${(\sqrt[3]{x} + 10)}^{2}$ as $(\sqrt[3]{x} + 10) (\sqrt[3]{x} + 10)$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 ((\sqrt[3]{x} + 10) (\sqrt[3]{x} + 10))) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Expand $(\sqrt[3]{x} + 10) (\sqrt[3]{x} + 10)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 (\sqrt[3]{x} (\sqrt[3]{x} + 10) + 10 (\sqrt[3]{x} + 10))) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Apply the distributive property.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 10 + 10 (\sqrt[3]{x} + 10))) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Apply the distributive property.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 10 + 10 \sqrt[3]{x} + 10 \cdot 10)) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt[3]{x} \sqrt[3]{x}$ .

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

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

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 ({\sqrt[3]{x}}^{1 + 1} + \sqrt[3]{x} \cdot 10 + 10 \sqrt[3]{x} + 10 \cdot 10)) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Add $1$ and $1$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 ({\sqrt[3]{x}}^{2} + \sqrt[3]{x} \cdot 10 + 10 \sqrt[3]{x} + 10 \cdot 10)) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

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

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 (\sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot 10 + 10 \sqrt[3]{x} + 10 \cdot 10)) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Move $10$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 (\sqrt[3]{x^{2}} + 10 \cdot \sqrt[3]{x} + 10 \sqrt[3]{x} + 10 \cdot 10)) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Multiply $10$ by $10$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 (\sqrt[3]{x^{2}} + 10 \sqrt[3]{x} + 10 \sqrt[3]{x} + 100)) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Add $10 \sqrt[3]{x}$ and $10 \sqrt[3]{x}$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 (\sqrt[3]{x^{2}} + 20 \sqrt[3]{x} + 100)) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Apply the distributive property.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 \sqrt[3]{x^{2}} + 30 (20 \sqrt[3]{x}) + 30 \cdot 100) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Simplify.

Multiply $20$ by $30$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 \sqrt[3]{x^{2}} + 600 \sqrt[3]{x} + 30 \cdot 100) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Multiply $30$ by $100$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 \sqrt[3]{x^{2}} + 600 \sqrt[3]{x} + 3000) + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Apply the distributive property.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - (30 \sqrt[3]{x^{2}}) - (600 \sqrt[3]{x}) - 1 \cdot 3000 + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Simplify.

Multiply $30$ by $- 1$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - (600 \sqrt[3]{x}) - 1 \cdot 3000 + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Multiply $600$ by $- 1$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - 600 \sqrt[3]{x} - 1 \cdot 3000 + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Multiply $- 1$ by $3000$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - 600 \sqrt[3]{x} - 3000 + 75 (4 (\sqrt[3]{x} + 10)) - 1000$

Apply the distributive property.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - 600 \sqrt[3]{x} - 3000 + 75 (4 \sqrt[3]{x} + 4 \cdot 10) - 1000$

Multiply $4$ by $10$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - 600 \sqrt[3]{x} - 3000 + 75 (4 \sqrt[3]{x} + 40) - 1000$

Apply the distributive property.

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - 600 \sqrt[3]{x} - 3000 + 75 (4 \sqrt[3]{x}) + 75 \cdot 40 - 1000$

Multiply $4$ by $75$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - 600 \sqrt[3]{x} - 3000 + 300 \sqrt[3]{x} + 75 \cdot 40 - 1000$

Multiply $75$ by $40$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - 600 \sqrt[3]{x} - 3000 + 300 \sqrt[3]{x} + 3000 - 1000$

Simplify by adding terms.

Combine the opposite terms in $x + 30 \sqrt[3]{x^{2}} + 300 \sqrt[3]{x} + 1000 - 30 \sqrt[3]{x^{2}} - 600 \sqrt[3]{x} - 3000 + 300 \sqrt[3]{x} + 3000 - 1000$ .

Subtract $30 \sqrt[3]{x^{2}}$ from $30 \sqrt[3]{x^{2}}$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 0 + 300 \sqrt[3]{x} + 1000 - 600 \sqrt[3]{x} - 3000 + 300 \sqrt[3]{x} + 3000 - 1000$

Add $x$ and $0$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 300 \sqrt[3]{x} + 1000 - 600 \sqrt[3]{x} - 3000 + 300 \sqrt[3]{x} + 3000 - 1000$

Add $- 3000$ and $3000$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 300 \sqrt[3]{x} + 1000 - 600 \sqrt[3]{x} + 0 + 300 \sqrt[3]{x} - 1000$

Add $x + 300 \sqrt[3]{x} + 1000 - 600 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 300 \sqrt[3]{x} + 1000 - 600 \sqrt[3]{x} + 300 \sqrt[3]{x} - 1000$

Subtract $1000$ from $1000$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 300 \sqrt[3]{x} + 0 - 600 \sqrt[3]{x} + 300 \sqrt[3]{x}$

Add $x + 300 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 300 \sqrt[3]{x} - 600 \sqrt[3]{x} + 300 \sqrt[3]{x}$

Subtract $600 \sqrt[3]{x}$ from $300 \sqrt[3]{x}$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x - 300 \sqrt[3]{x} + 300 \sqrt[3]{x}$

Combine the opposite terms in $x - 300 \sqrt[3]{x} + 300 \sqrt[3]{x}$ .

Add $- 300 \sqrt[3]{x}$ and $300 \sqrt[3]{x}$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (4 (\sqrt[3]{x} + 10)) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000} + 10)$

Remove parentheses.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000} + 10)$

Simplify each term.

To write $- \frac{15 x^{2}}{8}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3}}{64} - \frac{15 x^{2}}{8} \cdot \frac{8}{8} + 75 x - 1000} + 10)$

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{15 x^{2}}{8}$ by $\frac{8}{8}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3}}{64} - \frac{15 x^{2} \cdot 8}{8 \cdot 8} + 75 x - 1000} + 10)$

Multiply $8$ by $8$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3}}{64} - \frac{15 x^{2} \cdot 8}{64} + 75 x - 1000} + 10)$

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3} - 15 x^{2} \cdot 8}{64} + 75 x - 1000} + 10)$

Simplify the numerator.

Factor $x^{2}$ out of $x^{3} - 15 x^{2} \cdot 8$ .

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

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{2} x - 15 x^{2} \cdot 8}{64} + 75 x - 1000} + 10)$

Factor $x^{2}$ out of $- 15 x^{2} \cdot 8$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{2} x + x^{2} (- 15 \cdot 8)}{64} + 75 x - 1000} + 10)$

Factor $x^{2}$ out of $x^{2} x + x^{2} (- 15 \cdot 8)$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{2} (x - 15 \cdot 8)}{64} + 75 x - 1000} + 10)$

Multiply $- 15$ by $8$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{2} (x - 120)}{64} + 75 x - 1000} + 10)$

To write $75 x$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{2} (x - 120)}{64} + 75 x \cdot \frac{64}{64} - 1000} + 10)$

Combine $75 x$ and $\frac{64}{64}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{2} (x - 120)}{64} + \frac{75 x \cdot 64}{64} - 1000} + 10)$

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{2} (x - 120) + 75 x \cdot 64}{64} - 1000} + 10)$

Simplify the numerator.

Factor $x$ out of $x^{2} (x - 120) + 75 x \cdot 64$ .

Factor $x$ out of $x^{2} (x - 120)$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x (x - 120)) + 75 x \cdot 64}{64} - 1000} + 10)$

Factor $x$ out of $75 x \cdot 64$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x (x - 120)) + x (75 \cdot 64)}{64} - 1000} + 10)$

Factor $x$ out of $x (x (x - 120)) + x (75 \cdot 64)$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x (x - 120) + 75 \cdot 64)}{64} - 1000} + 10)$

Apply the distributive property.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x \cdot x + x \cdot - 120 + 75 \cdot 64)}{64} - 1000} + 10)$

Multiply $x$ by $x$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x^{2} + x \cdot - 120 + 75 \cdot 64)}{64} - 1000} + 10)$

Move $- 120$ to the left of $x$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x^{2} - 120 \cdot x + 75 \cdot 64)}{64} - 1000} + 10)$

Multiply $75$ by $64$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x^{2} - 120 x + 4800)}{64} - 1000} + 10)$

To write $- 1000$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x^{2} - 120 x + 4800)}{64} - 1000 \cdot \frac{64}{64}} + 10)$

Combine $- 1000$ and $\frac{64}{64}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x^{2} - 120 x + 4800)}{64} + \frac{- 1000 \cdot 64}{64}} + 10)$

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x (x^{2} - 120 x + 4800) - 1000 \cdot 64}{64}} + 10)$

Simplify the numerator.

Apply the distributive property.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x \cdot x^{2} + x (- 120 x) + x \cdot 4800 - 1000 \cdot 64}{64}} + 10)$

Simplify.

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

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

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

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x \cdot x^{2} + x (- 120 x) + x \cdot 4800 - 1000 \cdot 64}{64}} + 10)$

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

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{1 + 2} + x (- 120 x) + x \cdot 4800 - 1000 \cdot 64}{64}} + 10)$

Add $1$ and $2$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3} + x (- 120 x) + x \cdot 4800 - 1000 \cdot 64}{64}} + 10)$

Rewrite using the commutative property of multiplication.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3} - 120 x \cdot x + x \cdot 4800 - 1000 \cdot 64}{64}} + 10)$

Move $4800$ to the left of $x$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3} - 120 x \cdot x + 4800 \cdot x - 1000 \cdot 64}{64}} + 10)$

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

Move $x$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3} - 120 (x \cdot x) + 4800 \cdot x - 1000 \cdot 64}{64}} + 10)$

Multiply $x$ by $x$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3} - 120 x^{2} + 4800 \cdot x - 1000 \cdot 64}{64}} + 10)$

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3} - 120 x^{2} + 4800 x - 1000 \cdot 64}{64}} + 10)$

Multiply $- 1000$ by $64$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{x^{3} - 120 x^{2} + 4800 x - 64000}{64}} + 10)$

Rewrite $x^{3} - 120 x^{2} + 4800 x - 64000$ in a factored form.

Factor $x^{3} - 120 x^{2} + 4800 x - 64000$ 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.

$p = \pm 1, \pm 64000, \pm 2, \pm 32000, \pm 4, \pm 16000, \pm 5, \pm 12800, \pm 8, \pm 8000, \pm 10, \pm 6400, \pm 16, \pm 4000, \pm 20, \pm 3200, \pm 25, \pm 2560, \pm 32, \pm 2000, \pm 40, \pm 1600, \pm 50, \pm 1280, \pm 64, \pm 1000, \pm 80, \pm 800, \pm 100, \pm 640, \pm 125, \pm 512, \pm 128, \pm 500, \pm 160, \pm 400, \pm 200, \pm 320, \pm 250, \pm 256 q = \pm 1$

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

$\pm 1, \pm 64000, \pm 2, \pm 32000, \pm 4, \pm 16000, \pm 5, \pm 12800, \pm 8, \pm 8000, \pm 10, \pm 6400, \pm 16, \pm 4000, \pm 20, \pm 3200, \pm 25, \pm 2560, \pm 32, \pm 2000, \pm 40, \pm 1600, \pm 50, \pm 1280, \pm 64, \pm 1000, \pm 80, \pm 800, \pm 100, \pm 640, \pm 125, \pm 512, \pm 128, \pm 500, \pm 160, \pm 400, \pm 200, \pm 320, \pm 250, \pm 256$

Substitute $40$ and simplify the expression. In this case, the expression is equal to $0$ so $40$ is a root of the polynomial.

Substitute $40$ into the polynomial.

$40^{3} - 120 \cdot 40^{2} + 4800 \cdot 40 - 64000$

Raise $40$ to the power of $3$ .

$64000 - 120 \cdot 40^{2} + 4800 \cdot 40 - 64000$

Raise $40$ to the power of $2$ .

$64000 - 120 \cdot 1600 + 4800 \cdot 40 - 64000$

Multiply $- 120$ by $1600$ .

$64000 - 192000 + 4800 \cdot 40 - 64000$

Subtract $192000$ from $64000$ .

$- 128000 + 4800 \cdot 40 - 64000$

Multiply $4800$ by $40$ .

$- 128000 + 192000 - 64000$

Add $- 128000$ and $192000$ .

$64000 - 64000$

Subtract $64000$ from $64000$ .

$0$

Since $40$ is a known root, divide the polynomial by $x - 40$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 120 x^{2} + 4800 x - 64000}{x - 40}$

Divide $x^{3} - 120 x^{2} + 4800 x - 64000$ by $x - 40$ .

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

$x$

$40$

$x^{3}$

$120 x^{2}$

$4800 x$

$64000$

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

					$x^{2}$
	$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			+	$x^{3}$	-	$40 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 40 x^{2}$

				$x^{2}$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$

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

				$x^{2}$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$

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

				$x^{2}$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$

Divide the highest order term in the dividend $- 80 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$80 x$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$80 x$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$
					-	$80 x^{2}$	+	$3200 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 80 x^{2} + 3200 x$

				$x^{2}$	-	$80 x$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$
					+	$80 x^{2}$	-	$3200 x$

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

				$x^{2}$	-	$80 x$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$
					+	$80 x^{2}$	-	$3200 x$
							+	$1600 x$

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

				$x^{2}$	-	$80 x$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$
					+	$80 x^{2}$	-	$3200 x$
							+	$1600 x$	-	$64000$

Divide the highest order term in the dividend $1600 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$80 x$	+	$1600$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$
					+	$80 x^{2}$	-	$3200 x$
							+	$1600 x$	-	$64000$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$80 x$	+	$1600$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$
					+	$80 x^{2}$	-	$3200 x$
							+	$1600 x$	-	$64000$
							+	$1600 x$	-	$64000$

The expression needs to be subtracted from the dividend, so change all the signs in $1600 x - 64000$

				$x^{2}$	-	$80 x$	+	$1600$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$
					+	$80 x^{2}$	-	$3200 x$
							+	$1600 x$	-	$64000$
							-	$1600 x$	+	$64000$

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

				$x^{2}$	-	$80 x$	+	$1600$
$x$	-	$40$		$x^{3}$	-	$120 x^{2}$	+	$4800 x$	-	$64000$
			-	$x^{3}$	+	$40 x^{2}$
					-	$80 x^{2}$	+	$4800 x$
					+	$80 x^{2}$	-	$3200 x$
							+	$1600 x$	-	$64000$
							-	$1600 x$	+	$64000$
										$0$

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

$x^{2} - 80 x + 1600$

Write $x^{3} - 120 x^{2} + 4800 x - 64000$ as a set of factors.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{(x - 40) (x^{2} - 80 x + 1600)}{64}} + 10)$

Factor using the perfect square rule.

Rewrite $1600$ as $40^{2}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{(x - 40) (x^{2} - 80 x + 40^{2})}{64}} + 10)$

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

$80 x = 2 \cdot x \cdot 40$

Rewrite the polynomial.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{(x - 40) (x^{2} - 2 \cdot x \cdot 40 + 40^{2})}{64}} + 10)$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 40$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{(x - 40) {(x - 40)}^{2}}{64}} + 10)$

Combine like factors.

Raise $x - 40$ to the power of $1$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{(x - 40) {(x - 40)}^{2}}{64}} + 10)$

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

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{{(x - 40)}^{1 + 2}}{64}} + 10)$

Add $1$ and $2$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{{(x - 40)}^{3}}{64}} + 10)$

Rewrite $64$ as $4^{3}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{\frac{{(x - 40)}^{3}}{4^{3}}} + 10)$

Rewrite $\frac{{(x - 40)}^{3}}{4^{3}}$ as ${(\frac{x - 40}{4})}^{3}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\sqrt[3]{{(\frac{x - 40}{4})}^{3}} + 10)$

Pull terms out from under the radical, assuming real numbers.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\frac{x - 40}{4} + 10)$

To write $10$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\frac{x - 40}{4} + 10 \cdot \frac{4}{4})$

Simplify terms.

Combine $10$ and $\frac{4}{4}$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\frac{x - 40}{4} + \frac{10 \cdot 4}{4})$

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\frac{x - 40 + 10 \cdot 4}{4})$

Simplify the numerator.

Multiply $10$ by $4$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\frac{x - 40 + 40}{4})$

Add $- 40$ and $40$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\frac{x + 0}{4})$

Add $x$ and $0$ .

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\frac{x}{4})$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = 4 (\frac{x}{4})$

Rewrite the expression.

$f (\frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000$ is the inverse of $f (x) = 4 (\sqrt[3]{x} + 10)$ .

$f^{- 1} (x) = \frac{x^{3}}{64} - \frac{15 x^{2}}{8} + 75 x - 1000$

Find the Inverse 4( cube root of x+10)

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