Algebra Examples

$y = 3 \sqrt[3]{x} + 9$

Step 1

Interchange the variables.

$x = 3 \sqrt[3]{y} + 9$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $3 \sqrt[3]{y} + 9 = x$ .

$3 \sqrt[3]{y} + 9 = x$

Step 2.2

Subtract $9$ from both sides of the equation.

$3 \sqrt[3]{y} = x - 9$

Step 2.3

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

${(3 \sqrt[3]{y})}^{3} = {(x - 9)}^{3}$

Step 2.4

Simplify each side of the equation.

Tap for more steps...

Step 2.4.1

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

${(3 y^{\frac{1}{3}})}^{3} = {(x - 9)}^{3}$

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

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

Tap for more steps...

Step 2.4.2.1.1

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

$3^{3} {(y^{\frac{1}{3}})}^{3} = {(x - 9)}^{3}$

Step 2.4.2.1.2

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

$27 {(y^{\frac{1}{3}})}^{3} = {(x - 9)}^{3}$

Step 2.4.2.1.3

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

Tap for more steps...

Step 2.4.2.1.3.1

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

$27 y^{\frac{1}{3} \cdot 3} = {(x - 9)}^{3}$

Step 2.4.2.1.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.4.2.1.3.2.1

Cancel the common factor.

$27 y^{\frac{1}{3} \cdot 3} = {(x - 9)}^{3}$

Step 2.4.2.1.3.2.2

Rewrite the expression.

$27 y^{1} = {(x - 9)}^{3}$

Step 2.4.2.1.4

Simplify.

$27 y = {(x - 9)}^{3}$

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Simplify ${(x - 9)}^{3}$ .

Tap for more steps...

Step 2.4.3.1.1

Use the Binomial Theorem.

$27 y = x^{3} + 3 x^{2} \cdot - 9 + 3 x {(- 9)}^{2} + {(- 9)}^{3}$

Step 2.4.3.1.2

Simplify each term.

Tap for more steps...

Step 2.4.3.1.2.1

Multiply $- 9$ by $3$ .

$27 y = x^{3} - 27 x^{2} + 3 x {(- 9)}^{2} + {(- 9)}^{3}$

Step 2.4.3.1.2.2

Raise $- 9$ to the power of $2$ .

$27 y = x^{3} - 27 x^{2} + 3 x \cdot 81 + {(- 9)}^{3}$

Step 2.4.3.1.2.3

Multiply $81$ by $3$ .

$27 y = x^{3} - 27 x^{2} + 243 x + {(- 9)}^{3}$

Step 2.4.3.1.2.4

Raise $- 9$ to the power of $3$ .

$27 y = x^{3} - 27 x^{2} + 243 x - 729$

Step 2.5

Divide each term in $27 y = x^{3} - 27 x^{2} + 243 x - 729$ by $27$ and simplify.

Tap for more steps...

Step 2.5.1

Divide each term in $27 y = x^{3} - 27 x^{2} + 243 x - 729$ by $27$ .

$\frac{27 y}{27} = \frac{x^{3}}{27} + \frac{- 27 x^{2}}{27} + \frac{243 x}{27} + \frac{- 729}{27}$

Step 2.5.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.1

Cancel the common factor of $27$ .

Tap for more steps...

Step 2.5.2.1.1

Cancel the common factor.

$\frac{27 y}{27} = \frac{x^{3}}{27} + \frac{- 27 x^{2}}{27} + \frac{243 x}{27} + \frac{- 729}{27}$

Step 2.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{3}}{27} + \frac{- 27 x^{2}}{27} + \frac{243 x}{27} + \frac{- 729}{27}$

Step 2.5.3

Simplify the right side.

Tap for more steps...

Step 2.5.3.1

Simplify each term.

Tap for more steps...

Step 2.5.3.1.1

Cancel the common factor of $- 27$ and $27$ .

Tap for more steps...

Step 2.5.3.1.1.1

Factor $27$ out of $- 27 x^{2}$ .

$y = \frac{x^{3}}{27} + \frac{27 (- x^{2})}{27} + \frac{243 x}{27} + \frac{- 729}{27}$

Step 2.5.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 2.5.3.1.1.2.1

Factor $27$ out of $27$ .

$y = \frac{x^{3}}{27} + \frac{27 (- x^{2})}{27 (1)} + \frac{243 x}{27} + \frac{- 729}{27}$

Step 2.5.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{3}}{27} + \frac{27 (- x^{2})}{27 \cdot 1} + \frac{243 x}{27} + \frac{- 729}{27}$

Step 2.5.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{3}}{27} + \frac{- x^{2}}{1} + \frac{243 x}{27} + \frac{- 729}{27}$

Step 2.5.3.1.1.2.4

Divide $- x^{2}$ by $1$ .

$y = \frac{x^{3}}{27} - x^{2} + \frac{243 x}{27} + \frac{- 729}{27}$

Step 2.5.3.1.2

Cancel the common factor of $243$ and $27$ .

Tap for more steps...

Step 2.5.3.1.2.1

Factor $27$ out of $243 x$ .

$y = \frac{x^{3}}{27} - x^{2} + \frac{27 (9 x)}{27} + \frac{- 729}{27}$

Step 2.5.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 2.5.3.1.2.2.1

Factor $27$ out of $27$ .

$y = \frac{x^{3}}{27} - x^{2} + \frac{27 (9 x)}{27 (1)} + \frac{- 729}{27}$

Step 2.5.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{3}}{27} - x^{2} + \frac{27 (9 x)}{27 \cdot 1} + \frac{- 729}{27}$

Step 2.5.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{3}}{27} - x^{2} + \frac{9 x}{1} + \frac{- 729}{27}$

Step 2.5.3.1.2.2.4

Divide $9 x$ by $1$ .

$y = \frac{x^{3}}{27} - x^{2} + 9 x + \frac{- 729}{27}$

Step 2.5.3.1.3

Divide $- 729$ by $27$ .

$y = \frac{x^{3}}{27} - x^{2} + 9 x - 27$

Step 3

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

$f^{- 1} (x) = \frac{x^{3}}{27} - x^{2} + 9 x - 27$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{3}}{27} - x^{2} + 9 x - 27$ is the inverse of $f (x) = 3 \sqrt[3]{x} + 9$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (3 \sqrt[3]{x} + 9)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = \frac{{(3 \sqrt[3]{x} + 9)}^{3}}{27} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3

Simplify each term.

Tap for more steps...

Step 4.2.3.1

Simplify the numerator.

Tap for more steps...

Step 4.2.3.1.1

Factor $3$ out of $3 \sqrt[3]{x} + 9$ .

Tap for more steps...

Step 4.2.3.1.1.1

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = \frac{{(3 (\sqrt[3]{x}) + 9)}^{3}}{27} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.1.1.2

Factor $3$ out of $9$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = \frac{{(3 \sqrt[3]{x} + 3 \cdot 3)}^{3}}{27} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.1.1.3

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = \frac{{(3 (\sqrt[3]{x} + 3))}^{3}}{27} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.1.2

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = \frac{3^{3} {(\sqrt[3]{x} + 3)}^{3}}{27} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.1.3

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = \frac{27 {(\sqrt[3]{x} + 3)}^{3}}{27} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.2

Cancel the common factor of $27$ .

Tap for more steps...

Step 4.2.3.2.1

Cancel the common factor.

$f^{- 1} (3 \sqrt[3]{x} + 9) = \frac{27 {(\sqrt[3]{x} + 3)}^{3}}{27} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.2.2

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = {(\sqrt[3]{x} + 3)}^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.3

Use the Binomial Theorem.

$f^{- 1} (3 \sqrt[3]{x} + 9) = {\sqrt[3]{x}}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot 3 + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4

Simplify each term.

Tap for more steps...

Step 4.2.3.4.1

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

Tap for more steps...

Step 4.2.3.4.1.1

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = {(x^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot 3 + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.1.2

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x}}^{2} \cdot 3 + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.1.3

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot 3 + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.3.4.1.4.1

Cancel the common factor.

$f^{- 1} (3 \sqrt[3]{x} + 9) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot 3 + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.1.4.2

Rewrite the expression.

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 3 {\sqrt[3]{x}}^{2} \cdot 3 + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.1.5

Simplify.

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 3 {\sqrt[3]{x}}^{2} \cdot 3 + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.2

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 3 \sqrt[3]{x^{2}} \cdot 3 + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.3

Multiply $3$ by $3$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 3^{2} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.4

Multiply $3$ by $3^{2}$ by adding the exponents.

Tap for more steps...

Step 4.2.3.4.4.1

Move $3^{2}$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 3^{2} \cdot (3 \sqrt[3]{x}) + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.4.2

Multiply $3^{2}$ by $3$ .

Tap for more steps...

Step 4.2.3.4.4.2.1

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 3^{2} \cdot (3 \sqrt[3]{x}) + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.4.2.2

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 3^{2 + 1} \sqrt[3]{x} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.4.3

Add $2$ and $1$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 3^{3} \sqrt[3]{x} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.5

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 3^{3} - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.4.6

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - {(3 \sqrt[3]{x} + 9)}^{2} + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.5

Rewrite ${(3 \sqrt[3]{x} + 9)}^{2}$ as $(3 \sqrt[3]{x} + 9) (3 \sqrt[3]{x} + 9)$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - ((3 \sqrt[3]{x} + 9) (3 \sqrt[3]{x} + 9)) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.6

Expand $(3 \sqrt[3]{x} + 9) (3 \sqrt[3]{x} + 9)$ using the FOIL Method.

Tap for more steps...

Step 4.2.3.6.1

Apply the distributive property.

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (3 \sqrt[3]{x} (3 \sqrt[3]{x} + 9) + 9 (3 \sqrt[3]{x} + 9)) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.6.2

Apply the distributive property.

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (3 \sqrt[3]{x} (3 \sqrt[3]{x}) + 3 \sqrt[3]{x} \cdot 9 + 9 (3 \sqrt[3]{x} + 9)) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.6.3

Apply the distributive property.

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (3 \sqrt[3]{x} (3 \sqrt[3]{x}) + 3 \sqrt[3]{x} \cdot 9 + 9 (3 \sqrt[3]{x}) + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7

Simplify and combine like terms.

Tap for more steps...

Step 4.2.3.7.1

Simplify each term.

Tap for more steps...

Step 4.2.3.7.1.1

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

Tap for more steps...

Step 4.2.3.7.1.1.1

Multiply $3$ by $3$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 \sqrt[3]{x} \sqrt[3]{x} + 3 \sqrt[3]{x} \cdot 9 + 9 (3 \sqrt[3]{x}) + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.1.1.2

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 (\sqrt[3]{x} \sqrt[3]{x}) + 3 \sqrt[3]{x} \cdot 9 + 9 (3 \sqrt[3]{x}) + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.1.1.3

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 (\sqrt[3]{x} \sqrt[3]{x}) + 3 \sqrt[3]{x} \cdot 9 + 9 (3 \sqrt[3]{x}) + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.1.1.4

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

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 {\sqrt[3]{x}}^{1 + 1} + 3 \sqrt[3]{x} \cdot 9 + 9 (3 \sqrt[3]{x}) + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.1.1.5

Add $1$ and $1$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 {\sqrt[3]{x}}^{2} + 3 \sqrt[3]{x} \cdot 9 + 9 (3 \sqrt[3]{x}) + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.1.2

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 9 + 9 (3 \sqrt[3]{x}) + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.1.3

Multiply $9$ by $3$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 9 (3 \sqrt[3]{x}) + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.1.4

Multiply $3$ by $9$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 \sqrt[3]{x} + 9 \cdot 9) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.1.5

Multiply $9$ by $9$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 \sqrt[3]{x} + 81) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.7.2

Add $27 \sqrt[3]{x}$ and $27 \sqrt[3]{x}$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 \sqrt[3]{x^{2}} + 54 \sqrt[3]{x} + 81) + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.8

Apply the distributive property.

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - (9 \sqrt[3]{x^{2}}) - (54 \sqrt[3]{x}) - 1 \cdot 81 + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.9

Simplify.

Tap for more steps...

Step 4.2.3.9.1

Multiply $9$ by $- 1$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - 9 \sqrt[3]{x^{2}} - (54 \sqrt[3]{x}) - 1 \cdot 81 + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.9.2

Multiply $54$ by $- 1$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - 9 \sqrt[3]{x^{2}} - 54 \sqrt[3]{x} - 1 \cdot 81 + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.9.3

Multiply $- 1$ by $81$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - 9 \sqrt[3]{x^{2}} - 54 \sqrt[3]{x} - 81 + 9 (3 \sqrt[3]{x} + 9) - 27$

Step 4.2.3.10

Apply the distributive property.

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - 9 \sqrt[3]{x^{2}} - 54 \sqrt[3]{x} - 81 + 9 (3 \sqrt[3]{x}) + 9 \cdot 9 - 27$

Step 4.2.3.11

Multiply $3$ by $9$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - 9 \sqrt[3]{x^{2}} - 54 \sqrt[3]{x} - 81 + 27 \sqrt[3]{x} + 9 \cdot 9 - 27$

Step 4.2.3.12

Multiply $9$ by $9$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - 9 \sqrt[3]{x^{2}} - 54 \sqrt[3]{x} - 81 + 27 \sqrt[3]{x} + 81 - 27$

Step 4.2.4

Simplify by adding terms.

Tap for more steps...

Step 4.2.4.1

Combine the opposite terms in $x + 9 \sqrt[3]{x^{2}} + 27 \sqrt[3]{x} + 27 - 9 \sqrt[3]{x^{2}} - 54 \sqrt[3]{x} - 81 + 27 \sqrt[3]{x} + 81 - 27$ .

Tap for more steps...

Step 4.2.4.1.1

Subtract $9 \sqrt[3]{x^{2}}$ from $9 \sqrt[3]{x^{2}}$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 0 + 27 \sqrt[3]{x} + 27 - 54 \sqrt[3]{x} - 81 + 27 \sqrt[3]{x} + 81 - 27$

Step 4.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 27 \sqrt[3]{x} + 27 - 54 \sqrt[3]{x} - 81 + 27 \sqrt[3]{x} + 81 - 27$

Step 4.2.4.1.3

Add $- 81$ and $81$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 27 \sqrt[3]{x} + 27 - 54 \sqrt[3]{x} + 0 + 27 \sqrt[3]{x} - 27$

Step 4.2.4.1.4

Add $x + 27 \sqrt[3]{x} + 27 - 54 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 27 \sqrt[3]{x} + 27 - 54 \sqrt[3]{x} + 27 \sqrt[3]{x} - 27$

Step 4.2.4.1.5

Subtract $27$ from $27$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 27 \sqrt[3]{x} + 0 - 54 \sqrt[3]{x} + 27 \sqrt[3]{x}$

Step 4.2.4.1.6

Add $x + 27 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 27 \sqrt[3]{x} - 54 \sqrt[3]{x} + 27 \sqrt[3]{x}$

Step 4.2.4.2

Subtract $54 \sqrt[3]{x}$ from $27 \sqrt[3]{x}$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x - 27 \sqrt[3]{x} + 27 \sqrt[3]{x}$

Step 4.2.4.3

Combine the opposite terms in $x - 27 \sqrt[3]{x} + 27 \sqrt[3]{x}$ .

Tap for more steps...

Step 4.2.4.3.1

Add $- 27 \sqrt[3]{x}$ and $27 \sqrt[3]{x}$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x + 0$

Step 4.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (3 \sqrt[3]{x} + 9) = x$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x^{3}}{27} - x^{2} + 9 x - 27)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3}}{27} - x^{2} + 9 x - 27} + 9$

Step 4.3.3

Remove parentheses.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3}}{27} - x^{2} + 9 x - 27} + 9$

Step 4.3.4

Simplify each term.

Tap for more steps...

Step 4.3.4.1

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3}}{27} - x^{2} \cdot \frac{27}{27} + 9 x - 27} + 9$

Step 4.3.4.2

Combine $- x^{2}$ and $\frac{27}{27}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3}}{27} + \frac{- x^{2} \cdot 27}{27} + 9 x - 27} + 9$

Step 4.3.4.3

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3} - x^{2} \cdot 27}{27} + 9 x - 27} + 9$

Step 4.3.4.4

Simplify the numerator.

Tap for more steps...

Step 4.3.4.4.1

Factor $x^{2}$ out of $x^{3} - x^{2} \cdot 27$ .

Tap for more steps...

Step 4.3.4.4.1.1

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

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{2} x - x^{2} \cdot 27}{27} + 9 x - 27} + 9$

Step 4.3.4.4.1.2

Factor $x^{2}$ out of $- x^{2} \cdot 27$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{2} x + x^{2} (- 1 \cdot 27)}{27} + 9 x - 27} + 9$

Step 4.3.4.4.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} (- 1 \cdot 27)$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{2} (x - 1 \cdot 27)}{27} + 9 x - 27} + 9$

Step 4.3.4.4.2

Multiply $- 1$ by $27$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{2} (x - 27)}{27} + 9 x - 27} + 9$

Step 4.3.4.5

To write $9 x$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{2} (x - 27)}{27} + 9 x \cdot \frac{27}{27} - 27} + 9$

Step 4.3.4.6

Combine $9 x$ and $\frac{27}{27}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{2} (x - 27)}{27} + \frac{9 x \cdot 27}{27} - 27} + 9$

Step 4.3.4.7

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{2} (x - 27) + 9 x \cdot 27}{27} - 27} + 9$

Step 4.3.4.8

Simplify the numerator.

Tap for more steps...

Step 4.3.4.8.1

Factor $x$ out of $x^{2} (x - 27) + 9 x \cdot 27$ .

Tap for more steps...

Step 4.3.4.8.1.1

Factor $x$ out of $x^{2} (x - 27)$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x (x - 27)) + 9 x \cdot 27}{27} - 27} + 9$

Step 4.3.4.8.1.2

Factor $x$ out of $9 x \cdot 27$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x (x - 27)) + x (9 \cdot 27)}{27} - 27} + 9$

Step 4.3.4.8.1.3

Factor $x$ out of $x (x (x - 27)) + x (9 \cdot 27)$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x (x - 27) + 9 \cdot 27)}{27} - 27} + 9$

Step 4.3.4.8.2

Apply the distributive property.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x \cdot x + x \cdot - 27 + 9 \cdot 27)}{27} - 27} + 9$

Step 4.3.4.8.3

Multiply $x$ by $x$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x^{2} + x \cdot - 27 + 9 \cdot 27)}{27} - 27} + 9$

Step 4.3.4.8.4

Move $- 27$ to the left of $x$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x^{2} - 27 \cdot x + 9 \cdot 27)}{27} - 27} + 9$

Step 4.3.4.8.5

Multiply $9$ by $27$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x^{2} - 27 x + 243)}{27} - 27} + 9$

Step 4.3.4.9

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x^{2} - 27 x + 243)}{27} - 27 \cdot \frac{27}{27}} + 9$

Step 4.3.4.10

Combine $- 27$ and $\frac{27}{27}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x^{2} - 27 x + 243)}{27} + \frac{- 27 \cdot 27}{27}} + 9$

Step 4.3.4.11

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x (x^{2} - 27 x + 243) - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12

Simplify the numerator.

Tap for more steps...

Step 4.3.4.12.1

Apply the distributive property.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x \cdot x^{2} + x (- 27 x) + x \cdot 243 - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12.2

Simplify.

Tap for more steps...

Step 4.3.4.12.2.1

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

Tap for more steps...

Step 4.3.4.12.2.1.1

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

Tap for more steps...

Step 4.3.4.12.2.1.1.1

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

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x \cdot x^{2} + x (- 27 x) + x \cdot 243 - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12.2.1.1.2

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

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{1 + 2} + x (- 27 x) + x \cdot 243 - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12.2.1.2

Add $1$ and $2$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3} + x (- 27 x) + x \cdot 243 - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12.2.2

Rewrite using the commutative property of multiplication.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3} - 27 x \cdot x + x \cdot 243 - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12.2.3

Move $243$ to the left of $x$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3} - 27 x \cdot x + 243 \cdot x - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12.3

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

Tap for more steps...

Step 4.3.4.12.3.1

Move $x$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3} - 27 (x \cdot x) + 243 \cdot x - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12.3.2

Multiply $x$ by $x$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3} - 27 x^{2} + 243 \cdot x - 27 \cdot 27}{27}} + 9$

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3} - 27 x^{2} + 243 x - 27 \cdot 27}{27}} + 9$

Step 4.3.4.12.4

Multiply $- 27$ by $27$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{x^{3} - 27 x^{2} + 243 x - 729}{27}} + 9$

Step 4.3.4.12.5

Rewrite $x^{3} - 27 x^{2} + 243 x - 729$ in a factored form.

Tap for more steps...

Step 4.3.4.12.5.1

Factor $x^{3} - 27 x^{2} + 243 x - 729$ using the rational roots test.

Tap for more steps...

Step 4.3.4.12.5.1.1

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.

$p = \pm 1, \pm 729, \pm 3, \pm 243, \pm 9, \pm 81, \pm 27$

$q = \pm 1$

Step 4.3.4.12.5.1.2

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

$\pm 1, \pm 729, \pm 3, \pm 243, \pm 9, \pm 81, \pm 27$

Step 4.3.4.12.5.1.3

Substitute $9$ and simplify the expression. In this case, the expression is equal to $0$ so $9$ is a root of the polynomial.

Tap for more steps...

Step 4.3.4.12.5.1.3.1

Substitute $9$ into the polynomial.

$9^{3} - 27 \cdot 9^{2} + 243 \cdot 9 - 729$

Step 4.3.4.12.5.1.3.2

Raise $9$ to the power of $3$ .

$729 - 27 \cdot 9^{2} + 243 \cdot 9 - 729$

Step 4.3.4.12.5.1.3.3

Raise $9$ to the power of $2$ .

$729 - 27 \cdot 81 + 243 \cdot 9 - 729$

Step 4.3.4.12.5.1.3.4

Multiply $- 27$ by $81$ .

$729 - 2187 + 243 \cdot 9 - 729$

Step 4.3.4.12.5.1.3.5

Subtract $2187$ from $729$ .

$- 1458 + 243 \cdot 9 - 729$

Step 4.3.4.12.5.1.3.6

Multiply $243$ by $9$ .

$- 1458 + 2187 - 729$

Step 4.3.4.12.5.1.3.7

Add $- 1458$ and $2187$ .

$729 - 729$

Step 4.3.4.12.5.1.3.8

Subtract $729$ from $729$ .

$0$

Step 4.3.4.12.5.1.4

Since $9$ is a known root, divide the polynomial by $x - 9$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 27 x^{2} + 243 x - 729}{x - 9}$

Step 4.3.4.12.5.1.5

Divide $x^{3} - 27 x^{2} + 243 x - 729$ by $x - 9$ .

Tap for more steps...

Step 4.3.4.12.5.1.5.1

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

$x$

$9$

$x^{3}$

$27 x^{2}$

$243 x$

$729$

Step 4.3.4.12.5.1.5.2

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

					$x^{2}$
	$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$

Step 4.3.4.12.5.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			+	$x^{3}$	-	$9 x^{2}$

Step 4.3.4.12.5.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 9 x^{2}$

				$x^{2}$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$

Step 4.3.4.12.5.1.5.5

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

				$x^{2}$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$

Step 4.3.4.12.5.1.5.6

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

				$x^{2}$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$

Step 4.3.4.12.5.1.5.7

Divide the highest order term in the dividend $- 18 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$

Step 4.3.4.12.5.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					-	$18 x^{2}$	+	$162 x$

Step 4.3.4.12.5.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{2} + 162 x$

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$

Step 4.3.4.12.5.1.5.10

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

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$

Step 4.3.4.12.5.1.5.11

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

				$x^{2}$	-	$18 x$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$

Step 4.3.4.12.5.1.5.12

Divide the highest order term in the dividend $81 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$18 x$	+	$81$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$

Step 4.3.4.12.5.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$18 x$	+	$81$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$
							+	$81 x$	-	$729$

Step 4.3.4.12.5.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $81 x - 729$

				$x^{2}$	-	$18 x$	+	$81$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$
							-	$81 x$	+	$729$

Step 4.3.4.12.5.1.5.15

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

				$x^{2}$	-	$18 x$	+	$81$
$x$	-	$9$		$x^{3}$	-	$27 x^{2}$	+	$243 x$	-	$729$
			-	$x^{3}$	+	$9 x^{2}$
					-	$18 x^{2}$	+	$243 x$
					+	$18 x^{2}$	-	$162 x$
							+	$81 x$	-	$729$
							-	$81 x$	+	$729$
										$0$

Step 4.3.4.12.5.1.5.16

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

$x^{2} - 18 x + 81$

Step 4.3.4.12.5.1.6

Write $x^{3} - 27 x^{2} + 243 x - 729$ as a set of factors.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{(x - 9) (x^{2} - 18 x + 81)}{27}} + 9$

Step 4.3.4.12.5.2

Factor using the perfect square rule.

Tap for more steps...

Step 4.3.4.12.5.2.1

Rewrite $81$ as $9^{2}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{(x - 9) (x^{2} - 18 x + 9^{2})}{27}} + 9$

Step 4.3.4.12.5.2.2

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

$18 x = 2 \cdot x \cdot 9$

Step 4.3.4.12.5.2.3

Rewrite the polynomial.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{(x - 9) (x^{2} - 2 \cdot x \cdot 9 + 9^{2})}{27}} + 9$

Step 4.3.4.12.5.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 9$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{(x - 9) {(x - 9)}^{2}}{27}} + 9$

Step 4.3.4.12.5.3

Combine like factors.

Tap for more steps...

Step 4.3.4.12.5.3.1

Raise $x - 9$ to the power of $1$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{(x - 9) {(x - 9)}^{2}}{27}} + 9$

Step 4.3.4.12.5.3.2

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

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{{(x - 9)}^{1 + 2}}{27}} + 9$

Step 4.3.4.12.5.3.3

Add $1$ and $2$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{{(x - 9)}^{3}}{27}} + 9$

Step 4.3.4.13

Rewrite $27$ as $3^{3}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{\frac{{(x - 9)}^{3}}{3^{3}}} + 9$

Step 4.3.4.14

Rewrite $\frac{{(x - 9)}^{3}}{3^{3}}$ as ${(\frac{x - 9}{3})}^{3}$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 \sqrt[3]{{(\frac{x - 9}{3})}^{3}} + 9$

Step 4.3.4.15

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

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 (\frac{x - 9}{3}) + 9$

Step 4.3.4.16

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.4.16.1

Cancel the common factor.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = 3 (\frac{x - 9}{3}) + 9$

Step 4.3.4.16.2

Rewrite the expression.

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = x - 9 + 9$

Step 4.3.5

Combine the opposite terms in $x - 9 + 9$ .

Tap for more steps...

Step 4.3.5.1

Add $- 9$ and $9$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = x + 0$

Step 4.3.5.2

Add $x$ and $0$ .

$f (\frac{x^{3}}{27} - x^{2} + 9 x - 27) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{27} - x^{2} + 9 x - 27$ is the inverse of $f (x) = 3 \sqrt[3]{x} + 9$ .

$f^{- 1} (x) = \frac{x^{3}}{27} - x^{2} + 9 x - 27$