Find the Inverse cube root of x+11

$\sqrt[3]{x} + 11$

Step 1

Interchange the variables.

$x = \sqrt[3]{y} + 11$

Step 2

Solve for $y$ .

Rewrite the equation as $\sqrt[3]{y} + 11 = x$ .

$\sqrt[3]{y} + 11 = x$

Subtract $11$ from both sides of the equation.

$\sqrt[3]{y} = x - 11$

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

${\sqrt[3]{y}}^{3} = {(x - 11)}^{3}$

Simplify each side of the equation.

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

${(y^{\frac{1}{3}})}^{3} = {(x - 11)}^{3}$

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

$y^{\frac{1}{3} \cdot 3} = {(x - 11)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(x - 11)}^{3}$

Rewrite the expression.

$y^{1} = {(x - 11)}^{3}$

Simplify.

$y = {(x - 11)}^{3}$

Simplify the right side.

Simplify ${(x - 11)}^{3}$ .

Use the Binomial Theorem.

$y = x^{3} + 3 x^{2} \cdot - 11 + 3 x {(- 11)}^{2} + {(- 11)}^{3}$

Simplify each term.

Multiply $- 11$ by $3$ .

$y = x^{3} - 33 x^{2} + 3 x {(- 11)}^{2} + {(- 11)}^{3}$

Raise $- 11$ to the power of $2$ .

$y = x^{3} - 33 x^{2} + 3 x \cdot 121 + {(- 11)}^{3}$

Multiply $121$ by $3$ .

$y = x^{3} - 33 x^{2} + 363 x + {(- 11)}^{3}$

Raise $- 11$ to the power of $3$ .

$y = x^{3} - 33 x^{2} + 363 x - 1331$

Step 3

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

$f^{- 1} (x) = x^{3} - 33 x^{2} + 363 x - 1331$

Step 4

Verify if $f^{- 1} (x) = x^{3} - 33 x^{2} + 363 x - 1331$ is the inverse of $f (x) = \sqrt[3]{x} + 11$ .

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 $f^{- 1} (\sqrt[3]{x} + 11)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{x} + 11) = {(\sqrt[3]{x} + 11)}^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Simplify each term.

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{x} + 11) = {\sqrt[3]{x}}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot 11 + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

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 $x^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{x} + 11) = {(x^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot 11 + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

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

$f^{- 1} (\sqrt[3]{x} + 11) = x^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x}}^{2} \cdot 11 + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

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

$f^{- 1} (\sqrt[3]{x} + 11) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot 11 + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\sqrt[3]{x} + 11) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot 11 + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Rewrite the expression.

$f^{- 1} (\sqrt[3]{x} + 11) = x + 3 {\sqrt[3]{x}}^{2} \cdot 11 + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Simplify.

$f^{- 1} (\sqrt[3]{x} + 11) = x + 3 {\sqrt[3]{x}}^{2} \cdot 11 + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

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

$f^{- 1} (\sqrt[3]{x} + 11) = x + 3 \sqrt[3]{x^{2}} \cdot 11 + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Multiply $11$ by $3$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 11^{2} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Raise $11$ to the power of $2$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 121 + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Multiply $121$ by $3$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 11^{3} - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Raise $11$ to the power of $3$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 {(\sqrt[3]{x} + 11)}^{2} + 363 (\sqrt[3]{x} + 11) - 1331$

Rewrite ${(\sqrt[3]{x} + 11)}^{2}$ as $(\sqrt[3]{x} + 11) (\sqrt[3]{x} + 11)$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 ((\sqrt[3]{x} + 11) (\sqrt[3]{x} + 11)) + 363 (\sqrt[3]{x} + 11) - 1331$

Expand $(\sqrt[3]{x} + 11) (\sqrt[3]{x} + 11)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x} (\sqrt[3]{x} + 11) + 11 (\sqrt[3]{x} + 11)) + 363 (\sqrt[3]{x} + 11) - 1331$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 11 + 11 (\sqrt[3]{x} + 11)) + 363 (\sqrt[3]{x} + 11) - 1331$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 11 + 11 \sqrt[3]{x} + 11 \cdot 11) + 363 (\sqrt[3]{x} + 11) - 1331$

Simplify and combine like terms.

Simplify each term.

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

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

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 11 + 11 \sqrt[3]{x} + 11 \cdot 11) + 363 (\sqrt[3]{x} + 11) - 1331$

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

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 11 + 11 \sqrt[3]{x} + 11 \cdot 11) + 363 (\sqrt[3]{x} + 11) - 1331$

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

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 ({\sqrt[3]{x}}^{1 + 1} + \sqrt[3]{x} \cdot 11 + 11 \sqrt[3]{x} + 11 \cdot 11) + 363 (\sqrt[3]{x} + 11) - 1331$

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 ({\sqrt[3]{x}}^{2} + \sqrt[3]{x} \cdot 11 + 11 \sqrt[3]{x} + 11 \cdot 11) + 363 (\sqrt[3]{x} + 11) - 1331$

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

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot 11 + 11 \sqrt[3]{x} + 11 \cdot 11) + 363 (\sqrt[3]{x} + 11) - 1331$

Move $11$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x^{2}} + 11 \cdot \sqrt[3]{x} + 11 \sqrt[3]{x} + 11 \cdot 11) + 363 (\sqrt[3]{x} + 11) - 1331$

Multiply $11$ by $11$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x^{2}} + 11 \sqrt[3]{x} + 11 \sqrt[3]{x} + 121) + 363 (\sqrt[3]{x} + 11) - 1331$

Add $11 \sqrt[3]{x}$ and $11 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 (\sqrt[3]{x^{2}} + 22 \sqrt[3]{x} + 121) + 363 (\sqrt[3]{x} + 11) - 1331$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 \sqrt[3]{x^{2}} - 33 (22 \sqrt[3]{x}) - 33 \cdot 121 + 363 (\sqrt[3]{x} + 11) - 1331$

Simplify.

Multiply $22$ by $- 33$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 \sqrt[3]{x^{2}} - 726 \sqrt[3]{x} - 33 \cdot 121 + 363 (\sqrt[3]{x} + 11) - 1331$

Multiply $- 33$ by $121$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 \sqrt[3]{x^{2}} - 726 \sqrt[3]{x} - 3993 + 363 (\sqrt[3]{x} + 11) - 1331$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 \sqrt[3]{x^{2}} - 726 \sqrt[3]{x} - 3993 + 363 \sqrt[3]{x} + 363 \cdot 11 - 1331$

Multiply $363$ by $11$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 \sqrt[3]{x^{2}} - 726 \sqrt[3]{x} - 3993 + 363 \sqrt[3]{x} + 3993 - 1331$

Simplify by adding terms.

Combine the opposite terms in $x + 33 \sqrt[3]{x^{2}} + 363 \sqrt[3]{x} + 1331 - 33 \sqrt[3]{x^{2}} - 726 \sqrt[3]{x} - 3993 + 363 \sqrt[3]{x} + 3993 - 1331$ .

Subtract $33 \sqrt[3]{x^{2}}$ from $33 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 0 + 363 \sqrt[3]{x} + 1331 - 726 \sqrt[3]{x} - 3993 + 363 \sqrt[3]{x} + 3993 - 1331$

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 363 \sqrt[3]{x} + 1331 - 726 \sqrt[3]{x} - 3993 + 363 \sqrt[3]{x} + 3993 - 1331$

Add $- 3993$ and $3993$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 363 \sqrt[3]{x} + 1331 - 726 \sqrt[3]{x} + 0 + 363 \sqrt[3]{x} - 1331$

Add $x + 363 \sqrt[3]{x} + 1331 - 726 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 363 \sqrt[3]{x} + 1331 - 726 \sqrt[3]{x} + 363 \sqrt[3]{x} - 1331$

Subtract $1331$ from $1331$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 363 \sqrt[3]{x} + 0 - 726 \sqrt[3]{x} + 363 \sqrt[3]{x}$

Add $x + 363 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 363 \sqrt[3]{x} - 726 \sqrt[3]{x} + 363 \sqrt[3]{x}$

Subtract $726 \sqrt[3]{x}$ from $363 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x - 363 \sqrt[3]{x} + 363 \sqrt[3]{x}$

Combine the opposite terms in $x - 363 \sqrt[3]{x} + 363 \sqrt[3]{x}$ .

Add $- 363 \sqrt[3]{x}$ and $363 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x} + 11) = x$

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

Set up the composite result function.

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

Evaluate $f (x^{3} - 33 x^{2} + 363 x - 1331)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} - 33 x^{2} + 363 x - 1331) = \sqrt[3]{x^{3} - 33 x^{2} + 363 x - 1331} + 11$

Remove parentheses.

$f (x^{3} - 33 x^{2} + 363 x - 1331) = \sqrt[3]{x^{3} - 33 x^{2} + 363 x - 1331} + 11$

Simplify each term.

Make each term match the terms from the binomial theorem formula.

$f (x^{3} - 33 x^{2} + 363 x - 1331) = \sqrt[3]{x^{3} - 3 \cdot (11 x^{2}) + 3 \cdot (11^{2} x) - 1 \cdot 11^{3}} + 11$

Factor $x^{3} - 3 \cdot 11 x^{2} + 3 \cdot 11^{2} x - 1 \cdot 11^{3}$ using the binomial theorem.

$f (x^{3} - 33 x^{2} + 363 x - 1331) = \sqrt[3]{{(x - 11)}^{3}} + 11$

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

$f (x^{3} - 33 x^{2} + 363 x - 1331) = x - 11 + 11$

Combine the opposite terms in $x - 11 + 11$ .

Add $- 11$ and $11$ .

$f (x^{3} - 33 x^{2} + 363 x - 1331) = x + 0$

Add $x$ and $0$ .

$f (x^{3} - 33 x^{2} + 363 x - 1331) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} - 33 x^{2} + 363 x - 1331$ is the inverse of $f (x) = \sqrt[3]{x} + 11$ .

$f^{- 1} (x) = x^{3} - 33 x^{2} + 363 x - 1331$

Find the Inverse cube root of x+11

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