Precalculus Examples

$f (x) = \sqrt[5]{x} - 6$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \sqrt[5]{x} - 6$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[5]{y} - 6 = x$ .

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

Step 3.2

Add $6$ to both sides of the equation.

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

Step 3.3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{y}}^{5} = {(x + 6)}^{5}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y}$ as $y^{\frac{1}{5}}$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${(y^{\frac{1}{5}})}^{5}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${(y^{\frac{1}{5}})}^{5}$ .

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Step 3.4.2.1.1.1

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

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

Step 3.4.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(x + 6)}^{5}$

Step 3.4.2.1.2

Simplify.

$y = {(x + 6)}^{5}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x + 6)}^{5}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

$y = x^{5} + 5 x^{4} \cdot 6 + 10 x^{3} \cdot 6^{2} + 10 x^{2} \cdot 6^{3} + 5 x \cdot 6^{4} + 6^{5}$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Multiply $6$ by $5$ .

$y = x^{5} + 30 x^{4} + 10 x^{3} \cdot 6^{2} + 10 x^{2} \cdot 6^{3} + 5 x \cdot 6^{4} + 6^{5}$

Step 3.4.3.1.2.2

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

$y = x^{5} + 30 x^{4} + 10 x^{3} \cdot 36 + 10 x^{2} \cdot 6^{3} + 5 x \cdot 6^{4} + 6^{5}$

Step 3.4.3.1.2.3

Multiply $36$ by $10$ .

$y = x^{5} + 30 x^{4} + 360 x^{3} + 10 x^{2} \cdot 6^{3} + 5 x \cdot 6^{4} + 6^{5}$

Step 3.4.3.1.2.4

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

$y = x^{5} + 30 x^{4} + 360 x^{3} + 10 x^{2} \cdot 216 + 5 x \cdot 6^{4} + 6^{5}$

Step 3.4.3.1.2.5

Multiply $216$ by $10$ .

$y = x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 5 x \cdot 6^{4} + 6^{5}$

Step 3.4.3.1.2.6

Raise $6$ to the power of $4$ .

$y = x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 5 x \cdot 1296 + 6^{5}$

Step 3.4.3.1.2.7

Multiply $1296$ by $5$ .

$y = x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 6^{5}$

Step 3.4.3.1.2.8

Raise $6$ to the power of $5$ .

$y = x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776$

Step 4

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

$f^{- 1} (x) = x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776$

Step 5

Verify if $f^{- 1} (x) = x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776$ is the inverse of $f (x) = \sqrt[5]{x} - 6$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

$f^{- 1} (\sqrt[5]{x} - 6) = {(\sqrt[5]{x} - 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (\sqrt[5]{x} - 6) = {\sqrt[5]{x}}^{5} + 5 {\sqrt[5]{x}}^{4} \cdot - 6 + 10 {\sqrt[5]{x}}^{3} {(- 6)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

Rewrite ${\sqrt[5]{x}}^{5}$ as $x$ .

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Step 5.2.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = {(x^{\frac{1}{5}})}^{5} + 5 {\sqrt[5]{x}}^{4} \cdot - 6 + 10 {\sqrt[5]{x}}^{3} {(- 6)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.1.2

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

$f^{- 1} (\sqrt[5]{x} - 6) = x^{\frac{1}{5} \cdot 5} + 5 {\sqrt[5]{x}}^{4} \cdot - 6 + 10 {\sqrt[5]{x}}^{3} {(- 6)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.1.3

Combine $\frac{1}{5}$ and $5$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x^{\frac{5}{5}} + 5 {\sqrt[5]{x}}^{4} \cdot - 6 + 10 {\sqrt[5]{x}}^{3} {(- 6)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.1.4

Cancel the common factor of $5$ .

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Step 5.2.3.2.1.4.1

Cancel the common factor.

Step 5.2.3.2.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[5]{x} - 6) = x + 5 {\sqrt[5]{x}}^{4} \cdot - 6 + 10 {\sqrt[5]{x}}^{3} {(- 6)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.1.5

Simplify.

Step 5.2.3.2.2

Rewrite ${\sqrt[5]{x}}^{4}$ as $\sqrt[5]{x^{4}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 5 \sqrt[5]{x^{4}} \cdot - 6 + 10 {\sqrt[5]{x}}^{3} {(- 6)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.3

Multiply $- 6$ by $5$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 10 {\sqrt[5]{x}}^{3} {(- 6)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.4

Rewrite ${\sqrt[5]{x}}^{3}$ as $\sqrt[5]{x^{3}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 10 \sqrt[5]{x^{3}} {(- 6)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.5

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

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 10 \sqrt[5]{x^{3}} \cdot 36 + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.6

Multiply $36$ by $10$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} + 10 {\sqrt[5]{x}}^{2} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.7

Rewrite ${\sqrt[5]{x}}^{2}$ as $\sqrt[5]{x^{2}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} + 10 \sqrt[5]{x^{2}} {(- 6)}^{3} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.8

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

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} + 10 \sqrt[5]{x^{2}} \cdot - 216 + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.9

Multiply $- 216$ by $10$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 5 \sqrt[5]{x} {(- 6)}^{4} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.10

Raise $- 6$ to the power of $4$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 5 \sqrt[5]{x} \cdot 1296 + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.11

Multiply $1296$ by $5$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} + {(- 6)}^{5} + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.2.12

Raise $- 6$ to the power of $5$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 {(\sqrt[5]{x} - 6)}^{4} + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.3

Use the Binomial Theorem.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 ({\sqrt[5]{x}}^{4} + 4 {\sqrt[5]{x}}^{3} \cdot - 6 + 6 {\sqrt[5]{x}}^{2} {(- 6)}^{2} + 4 \sqrt[5]{x} {(- 6)}^{3} + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4

Simplify each term.

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Step 5.2.3.4.1

Rewrite ${\sqrt[5]{x}}^{4}$ as $\sqrt[5]{x^{4}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} + 4 {\sqrt[5]{x}}^{3} \cdot - 6 + 6 {\sqrt[5]{x}}^{2} {(- 6)}^{2} + 4 \sqrt[5]{x} {(- 6)}^{3} + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4.2

Rewrite ${\sqrt[5]{x}}^{3}$ as $\sqrt[5]{x^{3}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} + 4 \sqrt[5]{x^{3}} \cdot - 6 + 6 {\sqrt[5]{x}}^{2} {(- 6)}^{2} + 4 \sqrt[5]{x} {(- 6)}^{3} + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4.3

Multiply $- 6$ by $4$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} - 24 \sqrt[5]{x^{3}} + 6 {\sqrt[5]{x}}^{2} {(- 6)}^{2} + 4 \sqrt[5]{x} {(- 6)}^{3} + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4.4

Rewrite ${\sqrt[5]{x}}^{2}$ as $\sqrt[5]{x^{2}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} - 24 \sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} {(- 6)}^{2} + 4 \sqrt[5]{x} {(- 6)}^{3} + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4.5

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

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} - 24 \sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} \cdot 36 + 4 \sqrt[5]{x} {(- 6)}^{3} + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4.6

Multiply $36$ by $6$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} - 24 \sqrt[5]{x^{3}} + 216 \sqrt[5]{x^{2}} + 4 \sqrt[5]{x} {(- 6)}^{3} + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4.7

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

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} - 24 \sqrt[5]{x^{3}} + 216 \sqrt[5]{x^{2}} + 4 \sqrt[5]{x} \cdot - 216 + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4.8

Multiply $- 216$ by $4$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} - 24 \sqrt[5]{x^{3}} + 216 \sqrt[5]{x^{2}} - 864 \sqrt[5]{x} + {(- 6)}^{4}) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.4.9

Raise $- 6$ to the power of $4$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 (\sqrt[5]{x^{4}} - 24 \sqrt[5]{x^{3}} + 216 \sqrt[5]{x^{2}} - 864 \sqrt[5]{x} + 1296) + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.5

Apply the distributive property.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} + 30 (- 24 \sqrt[5]{x^{3}}) + 30 (216 \sqrt[5]{x^{2}}) + 30 (- 864 \sqrt[5]{x}) + 30 \cdot 1296 + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.6

Simplify.

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Step 5.2.3.6.1

Multiply $- 24$ by $30$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 30 (216 \sqrt[5]{x^{2}}) + 30 (- 864 \sqrt[5]{x}) + 30 \cdot 1296 + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.6.2

Multiply $216$ by $30$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} + 30 (- 864 \sqrt[5]{x}) + 30 \cdot 1296 + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.6.3

Multiply $- 864$ by $30$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 30 \cdot 1296 + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.6.4

Multiply $30$ by $1296$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 {(\sqrt[5]{x} - 6)}^{3} + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.7

Use the Binomial Theorem.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 ({\sqrt[5]{x}}^{3} + 3 {\sqrt[5]{x}}^{2} \cdot - 6 + 3 \sqrt[5]{x} {(- 6)}^{2} + {(- 6)}^{3}) + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.8

Simplify each term.

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Step 5.2.3.8.1

Rewrite ${\sqrt[5]{x}}^{3}$ as $\sqrt[5]{x^{3}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 (\sqrt[5]{x^{3}} + 3 {\sqrt[5]{x}}^{2} \cdot - 6 + 3 \sqrt[5]{x} {(- 6)}^{2} + {(- 6)}^{3}) + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.8.2

Rewrite ${\sqrt[5]{x}}^{2}$ as $\sqrt[5]{x^{2}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 (\sqrt[5]{x^{3}} + 3 \sqrt[5]{x^{2}} \cdot - 6 + 3 \sqrt[5]{x} {(- 6)}^{2} + {(- 6)}^{3}) + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.8.3

Multiply $- 6$ by $3$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 (\sqrt[5]{x^{3}} - 18 \sqrt[5]{x^{2}} + 3 \sqrt[5]{x} {(- 6)}^{2} + {(- 6)}^{3}) + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.8.4

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

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 (\sqrt[5]{x^{3}} - 18 \sqrt[5]{x^{2}} + 3 \sqrt[5]{x} \cdot 36 + {(- 6)}^{3}) + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.8.5

Multiply $36$ by $3$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 (\sqrt[5]{x^{3}} - 18 \sqrt[5]{x^{2}} + 108 \sqrt[5]{x} + {(- 6)}^{3}) + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.8.6

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

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 (\sqrt[5]{x^{3}} - 18 \sqrt[5]{x^{2}} + 108 \sqrt[5]{x} - 216) + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.9

Apply the distributive property.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} + 360 (- 18 \sqrt[5]{x^{2}}) + 360 (108 \sqrt[5]{x}) + 360 \cdot - 216 + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.10

Simplify.

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Step 5.2.3.10.1

Multiply $- 18$ by $360$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 360 (108 \sqrt[5]{x}) + 360 \cdot - 216 + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.10.2

Multiply $108$ by $360$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} + 360 \cdot - 216 + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.10.3

Multiply $360$ by $- 216$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 {(\sqrt[5]{x} - 6)}^{2} + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.11

Rewrite ${(\sqrt[5]{x} - 6)}^{2}$ as $(\sqrt[5]{x} - 6) (\sqrt[5]{x} - 6)$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 ((\sqrt[5]{x} - 6) (\sqrt[5]{x} - 6)) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.12

Expand $(\sqrt[5]{x} - 6) (\sqrt[5]{x} - 6)$ using the FOIL Method.

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Step 5.2.3.12.1

Apply the distributive property.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x} (\sqrt[5]{x} - 6) - 6 (\sqrt[5]{x} - 6)) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.12.2

Apply the distributive property.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot - 6 - 6 (\sqrt[5]{x} - 6)) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.12.3

Apply the distributive property.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot - 6 - 6 \sqrt[5]{x} - 6 \cdot - 6) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.13

Simplify and combine like terms.

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Step 5.2.3.13.1

Simplify each term.

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Step 5.2.3.13.1.1

Multiply $\sqrt[5]{x} \sqrt[5]{x}$ .

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Step 5.2.3.13.1.1.1

Raise $\sqrt[5]{x}$ to the power of $1$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot - 6 - 6 \sqrt[5]{x} - 6 \cdot - 6) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.13.1.1.2

Raise $\sqrt[5]{x}$ to the power of $1$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot - 6 - 6 \sqrt[5]{x} - 6 \cdot - 6) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.13.1.1.3

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

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 ({\sqrt[5]{x}}^{1 + 1} + \sqrt[5]{x} \cdot - 6 - 6 \sqrt[5]{x} - 6 \cdot - 6) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.13.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 ({\sqrt[5]{x}}^{2} + \sqrt[5]{x} \cdot - 6 - 6 \sqrt[5]{x} - 6 \cdot - 6) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.13.1.2

Rewrite ${\sqrt[5]{x}}^{2}$ as $\sqrt[5]{x^{2}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x^{2}} + \sqrt[5]{x} \cdot - 6 - 6 \sqrt[5]{x} - 6 \cdot - 6) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.13.1.3

Move $- 6$ to the left of $\sqrt[5]{x}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x^{2}} - 6 \cdot \sqrt[5]{x} - 6 \sqrt[5]{x} - 6 \cdot - 6) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.13.1.4

Multiply $- 6$ by $- 6$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x^{2}} - 6 \sqrt[5]{x} - 6 \sqrt[5]{x} + 36) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.13.2

Subtract $6 \sqrt[5]{x}$ from $- 6 \sqrt[5]{x}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 (\sqrt[5]{x^{2}} - 12 \sqrt[5]{x} + 36) + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.14

Apply the distributive property.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} + 2160 (- 12 \sqrt[5]{x}) + 2160 \cdot 36 + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.15

Simplify.

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Step 5.2.3.15.1

Multiply $- 12$ by $2160$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 2160 \cdot 36 + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.15.2

Multiply $2160$ by $36$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 77760 + 6480 (\sqrt[5]{x} - 6) + 7776$

Step 5.2.3.16

Apply the distributive property.

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} + 6480 \cdot - 6 + 7776$

Step 5.2.3.17

Multiply $6480$ by $- 6$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x - 30 \sqrt[5]{x^{4}} + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 + 30 \sqrt[5]{x^{4}} - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} - 38880 + 7776$ .

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Step 5.2.4.1.1

Add $- 30 \sqrt[5]{x^{4}}$ and $30 \sqrt[5]{x^{4}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 0 + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} - 6480 \sqrt[5]{x^{2}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4.1.3

Subtract $6480 \sqrt[5]{x^{2}}$ from $6480 \sqrt[5]{x^{2}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} + 0 - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4.1.4

Add $x + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}}$ and $0$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} - 2160 \sqrt[5]{x^{2}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 77760 + 2160 \sqrt[5]{x^{2}} - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4.1.5

Add $- 2160 \sqrt[5]{x^{2}}$ and $2160 \sqrt[5]{x^{2}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} + 0 + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 77760 - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4.1.6

Add $x + 360 \sqrt[5]{x^{3}}$ and $0$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 77760 - 25920 \sqrt[5]{x} + 77760 + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4.1.7

Add $- 77760$ and $77760$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} + 0 - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4.1.8

Add $x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 38880 + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x} - 38880 + 7776$

Step 5.2.4.1.9

Subtract $38880$ from $38880$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 0 + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x} + 7776$

Step 5.2.4.1.10

Add $x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 7776 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x} + 7776$

Step 5.2.4.1.11

Add $- 7776$ and $7776$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} + 0 - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x}$

Step 5.2.4.1.12

Add $x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 720 \sqrt[5]{x^{3}} - 25920 \sqrt[5]{x} + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x}$

Step 5.2.4.2

Subtract $720 \sqrt[5]{x^{3}}$ from $360 \sqrt[5]{x^{3}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x}$

Step 5.2.4.3

Combine the opposite terms in $x - 360 \sqrt[5]{x^{3}} + 6480 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 360 \sqrt[5]{x^{3}} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x}$ .

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Step 5.2.4.3.1

Add $- 360 \sqrt[5]{x^{3}}$ and $360 \sqrt[5]{x^{3}}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 0 + 6480 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x}$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 6480 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x}$

Step 5.2.4.4

Subtract $25920 \sqrt[5]{x}$ from $6480 \sqrt[5]{x}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 19440 \sqrt[5]{x} + 38880 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x}$

Step 5.2.4.5

Add $- 19440 \sqrt[5]{x}$ and $38880 \sqrt[5]{x}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x + 19440 \sqrt[5]{x} - 25920 \sqrt[5]{x} + 6480 \sqrt[5]{x}$

Step 5.2.4.6

Subtract $25920 \sqrt[5]{x}$ from $19440 \sqrt[5]{x}$ .

$f^{- 1} (\sqrt[5]{x} - 6) = x - 6480 \sqrt[5]{x} + 6480 \sqrt[5]{x}$

Step 5.2.4.7

Combine the opposite terms in $x - 6480 \sqrt[5]{x} + 6480 \sqrt[5]{x}$ .

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Step 5.2.4.7.1

Add $- 6480 \sqrt[5]{x}$ and $6480 \sqrt[5]{x}$ .

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

Step 5.2.4.7.2

Add $x$ and $0$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776) = \sqrt[5]{x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776} - 6$

Step 5.3.3

Remove parentheses.

$f (x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776) = \sqrt[5]{x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776} - 6$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

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

$f (x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776) = \sqrt[5]{x^{5} + 5 \cdot (6 x^{4}) + 10 \cdot (6^{2} x^{3}) + 10 \cdot (6^{3} x^{2}) + 5 \cdot (6^{4} x) + 6^{5}} - 6$

Step 5.3.4.2

Factor $x^{5} + 5 \cdot 6 x^{4} + 10 \cdot 6^{2} x^{3} + 10 \cdot 6^{3} x^{2} + 5 \cdot 6^{4} x + 6^{5}$ using the binomial theorem.

$f (x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776) = \sqrt[5]{{(x + 6)}^{5}} - 6$

Step 5.3.4.3

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

$f (x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776) = x + 6 - 6$

Step 5.3.5

Combine the opposite terms in $x + 6 - 6$ .

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Step 5.3.5.1

Subtract $6$ from $6$ .

$f (x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776$ is the inverse of $f (x) = \sqrt[5]{x} - 6$ .

$f^{- 1} (x) = x^{5} + 30 x^{4} + 360 x^{3} + 2160 x^{2} + 6480 x + 7776$