Precalculus Examples

$f (x) = \frac{\sqrt[5]{x} - 4}{8}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{\sqrt[5]{x} - 4}{8}$ as an equation.

$y = \frac{\sqrt[5]{x} - 4}{8}$

Step 2

Interchange the variables.

$x = \frac{\sqrt[5]{y} - 4}{8}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{\sqrt[5]{y} - 4}{8} = x$ .

$\frac{\sqrt[5]{y} - 4}{8} = x$

Step 3.2

Multiply both sides of the equation by $8$ .

$8 \frac{\sqrt[5]{y} - 4}{8} = 8 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $8$ .

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Step 3.3.1.1

Cancel the common factor.

$8 \frac{\sqrt[5]{y} - 4}{8} = 8 x$

Step 3.3.1.2

Rewrite the expression.

$\sqrt[5]{y} - 4 = 8 x$

Step 3.4

Add $4$ to both sides of the equation.

$\sqrt[5]{y} = 8 x + 4$

Step 3.5

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

${\sqrt[5]{y}}^{5} = {(8 x + 4)}^{5}$

Step 3.6

Simplify each side of the equation.

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Step 3.6.1

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

${(y^{\frac{1}{5}})}^{5} = {(8 x + 4)}^{5}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

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

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Step 3.6.2.1.1

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

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Step 3.6.2.1.1.1

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

$y^{\frac{1}{5} \cdot 5} = {(8 x + 4)}^{5}$

Step 3.6.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.6.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{5} \cdot 5} = {(8 x + 4)}^{5}$

Step 3.6.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(8 x + 4)}^{5}$

Step 3.6.2.1.2

Simplify.

$y = {(8 x + 4)}^{5}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Simplify ${(8 x + 4)}^{5}$ .

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Step 3.6.3.1.1

Use the Binomial Theorem.

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

Step 3.6.3.1.2

Simplify each term.

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Step 3.6.3.1.2.1

Apply the product rule to $8 x$ .

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

Step 3.6.3.1.2.2

Raise $8$ to the power of $5$ .

$y = 32768 x^{5} + 5 {(8 x)}^{4} \cdot 4 + 10 {(8 x)}^{3} \cdot 4^{2} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.3

Apply the product rule to $8 x$ .

$y = 32768 x^{5} + 5 (8^{4} x^{4}) \cdot 4 + 10 {(8 x)}^{3} \cdot 4^{2} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.4

Raise $8$ to the power of $4$ .

$y = 32768 x^{5} + 5 (4096 x^{4}) \cdot 4 + 10 {(8 x)}^{3} \cdot 4^{2} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.5

Multiply $4096$ by $5$ .

$y = 32768 x^{5} + 20480 x^{4} \cdot 4 + 10 {(8 x)}^{3} \cdot 4^{2} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.6

Multiply $4$ by $20480$ .

$y = 32768 x^{5} + 81920 x^{4} + 10 {(8 x)}^{3} \cdot 4^{2} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.7

Apply the product rule to $8 x$ .

$y = 32768 x^{5} + 81920 x^{4} + 10 (8^{3} x^{3}) \cdot 4^{2} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.8

Raise $8$ to the power of $3$ .

$y = 32768 x^{5} + 81920 x^{4} + 10 (512 x^{3}) \cdot 4^{2} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.9

Multiply $512$ by $10$ .

$y = 32768 x^{5} + 81920 x^{4} + 5120 x^{3} \cdot 4^{2} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.10

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

$y = 32768 x^{5} + 81920 x^{4} + 5120 x^{3} \cdot 16 + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.11

Multiply $16$ by $5120$ .

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 10 {(8 x)}^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.12

Apply the product rule to $8 x$ .

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 10 (8^{2} x^{2}) \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.13

Raise $8$ to the power of $2$ .

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 10 (64 x^{2}) \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.14

Multiply $64$ by $10$ .

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 640 x^{2} \cdot 4^{3} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.15

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

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 640 x^{2} \cdot 64 + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.16

Multiply $64$ by $640$ .

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 5 (8 x) \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.17

Multiply $8$ by $5$ .

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 40 x \cdot 4^{4} + 4^{5}$

Step 3.6.3.1.2.18

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

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 40 x \cdot 256 + 4^{5}$

Step 3.6.3.1.2.19

Multiply $256$ by $40$ .

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 4^{5}$

Step 3.6.3.1.2.20

Raise $4$ to the power of $5$ .

$y = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024$

Step 4

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

$f^{- 1} (x) = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024$

Step 5

Verify if $f^{- 1} (x) = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024$ is the inverse of $f (x) = \frac{\sqrt[5]{x} - 4}{8}$ .

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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 $f^{- 1} (\frac{\sqrt[5]{x} - 4}{8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = 32768 {(\frac{\sqrt[5]{x} - 4}{8})}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the product rule to $\frac{\sqrt[5]{x} - 4}{8}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = 32768 (\frac{{(\sqrt[5]{x} - 4)}^{5}}{8^{5}}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.2

Raise $8$ to the power of $5$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = 32768 (\frac{{(\sqrt[5]{x} - 4)}^{5}}{32768}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.3

Cancel the common factor of $32768$ .

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Step 5.2.3.3.1

Cancel the common factor.

Step 5.2.3.3.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = {(\sqrt[5]{x} - 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.4

Use the Binomial Theorem.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = {\sqrt[5]{x}}^{5} + 5 {\sqrt[5]{x}}^{4} \cdot - 4 + 10 {\sqrt[5]{x}}^{3} {(- 4)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5

Simplify each term.

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Step 5.2.3.5.1

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

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Step 5.2.3.5.1.1

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = {(x^{\frac{1}{5}})}^{5} + 5 {\sqrt[5]{x}}^{4} \cdot - 4 + 10 {\sqrt[5]{x}}^{3} {(- 4)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.1.2

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x^{\frac{1}{5} \cdot 5} + 5 {\sqrt[5]{x}}^{4} \cdot - 4 + 10 {\sqrt[5]{x}}^{3} {(- 4)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.1.3

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x^{\frac{5}{5}} + 5 {\sqrt[5]{x}}^{4} \cdot - 4 + 10 {\sqrt[5]{x}}^{3} {(- 4)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.1.4

Cancel the common factor of $5$ .

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Step 5.2.3.5.1.4.1

Cancel the common factor.

Step 5.2.3.5.1.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 5 {\sqrt[5]{x}}^{4} \cdot - 4 + 10 {\sqrt[5]{x}}^{3} {(- 4)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.1.5

Simplify.

Step 5.2.3.5.2

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 5 \sqrt[5]{x^{4}} \cdot - 4 + 10 {\sqrt[5]{x}}^{3} {(- 4)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.3

Multiply $- 4$ by $5$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 10 {\sqrt[5]{x}}^{3} {(- 4)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.4

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 10 \sqrt[5]{x^{3}} {(- 4)}^{2} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.5

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 10 \sqrt[5]{x^{3}} \cdot 16 + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.6

Multiply $16$ by $10$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} + 10 {\sqrt[5]{x}}^{2} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.7

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} + 10 \sqrt[5]{x^{2}} {(- 4)}^{3} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.8

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} + 10 \sqrt[5]{x^{2}} \cdot - 64 + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.9

Multiply $- 64$ by $10$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 5 \sqrt[5]{x} {(- 4)}^{4} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.10

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 5 \sqrt[5]{x} \cdot 256 + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.11

Multiply $256$ by $5$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} + {(- 4)}^{5} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.5.12

Raise $- 4$ to the power of $5$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.6

Apply the product rule to $\frac{\sqrt[5]{x} - 4}{8}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 81920 (\frac{{(\sqrt[5]{x} - 4)}^{4}}{8^{4}}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.7

Raise $8$ to the power of $4$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 81920 (\frac{{(\sqrt[5]{x} - 4)}^{4}}{4096}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.8

Cancel the common factor of $4096$ .

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Step 5.2.3.8.1

Factor $4096$ out of $81920$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 4096 (20) (\frac{{(\sqrt[5]{x} - 4)}^{4}}{4096}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.8.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 4096 \cdot (20 (\frac{{(\sqrt[5]{x} - 4)}^{4}}{4096})) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.8.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 {(\sqrt[5]{x} - 4)}^{4} + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.9

Use the Binomial Theorem.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 ({\sqrt[5]{x}}^{4} + 4 {\sqrt[5]{x}}^{3} \cdot - 4 + 6 {\sqrt[5]{x}}^{2} {(- 4)}^{2} + 4 \sqrt[5]{x} {(- 4)}^{3} + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10

Simplify each term.

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Step 5.2.3.10.1

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} + 4 {\sqrt[5]{x}}^{3} \cdot - 4 + 6 {\sqrt[5]{x}}^{2} {(- 4)}^{2} + 4 \sqrt[5]{x} {(- 4)}^{3} + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10.2

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} + 4 \sqrt[5]{x^{3}} \cdot - 4 + 6 {\sqrt[5]{x}}^{2} {(- 4)}^{2} + 4 \sqrt[5]{x} {(- 4)}^{3} + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10.3

Multiply $- 4$ by $4$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} - 16 \sqrt[5]{x^{3}} + 6 {\sqrt[5]{x}}^{2} {(- 4)}^{2} + 4 \sqrt[5]{x} {(- 4)}^{3} + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10.4

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} - 16 \sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} {(- 4)}^{2} + 4 \sqrt[5]{x} {(- 4)}^{3} + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10.5

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} - 16 \sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} \cdot 16 + 4 \sqrt[5]{x} {(- 4)}^{3} + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10.6

Multiply $16$ by $6$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} - 16 \sqrt[5]{x^{3}} + 96 \sqrt[5]{x^{2}} + 4 \sqrt[5]{x} {(- 4)}^{3} + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10.7

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} - 16 \sqrt[5]{x^{3}} + 96 \sqrt[5]{x^{2}} + 4 \sqrt[5]{x} \cdot - 64 + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10.8

Multiply $- 64$ by $4$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} - 16 \sqrt[5]{x^{3}} + 96 \sqrt[5]{x^{2}} - 256 \sqrt[5]{x} + {(- 4)}^{4}) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.10.9

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 (\sqrt[5]{x^{4}} - 16 \sqrt[5]{x^{3}} + 96 \sqrt[5]{x^{2}} - 256 \sqrt[5]{x} + 256) + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.11

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} + 20 (- 16 \sqrt[5]{x^{3}}) + 20 (96 \sqrt[5]{x^{2}}) + 20 (- 256 \sqrt[5]{x}) + 20 \cdot 256 + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.12

Simplify.

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Step 5.2.3.12.1

Multiply $- 16$ by $20$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 20 (96 \sqrt[5]{x^{2}}) + 20 (- 256 \sqrt[5]{x}) + 20 \cdot 256 + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.12.2

Multiply $96$ by $20$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} + 20 (- 256 \sqrt[5]{x}) + 20 \cdot 256 + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.12.3

Multiply $- 256$ by $20$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 20 \cdot 256 + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.12.4

Multiply $20$ by $256$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 81920 {(\frac{\sqrt[5]{x} - 4}{8})}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.13

Apply the product rule to $\frac{\sqrt[5]{x} - 4}{8}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 81920 (\frac{{(\sqrt[5]{x} - 4)}^{3}}{8^{3}}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.14

Raise $8$ to the power of $3$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 81920 (\frac{{(\sqrt[5]{x} - 4)}^{3}}{512}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.15

Cancel the common factor of $512$ .

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Step 5.2.3.15.1

Factor $512$ out of $81920$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 512 (160) (\frac{{(\sqrt[5]{x} - 4)}^{3}}{512}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.15.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 512 \cdot (160 (\frac{{(\sqrt[5]{x} - 4)}^{3}}{512})) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.15.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 {(\sqrt[5]{x} - 4)}^{3} + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.16

Use the Binomial Theorem.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 ({\sqrt[5]{x}}^{3} + 3 {\sqrt[5]{x}}^{2} \cdot - 4 + 3 \sqrt[5]{x} {(- 4)}^{2} + {(- 4)}^{3}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.17

Simplify each term.

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Step 5.2.3.17.1

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 (\sqrt[5]{x^{3}} + 3 {\sqrt[5]{x}}^{2} \cdot - 4 + 3 \sqrt[5]{x} {(- 4)}^{2} + {(- 4)}^{3}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.17.2

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 (\sqrt[5]{x^{3}} + 3 \sqrt[5]{x^{2}} \cdot - 4 + 3 \sqrt[5]{x} {(- 4)}^{2} + {(- 4)}^{3}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.17.3

Multiply $- 4$ by $3$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 (\sqrt[5]{x^{3}} - 12 \sqrt[5]{x^{2}} + 3 \sqrt[5]{x} {(- 4)}^{2} + {(- 4)}^{3}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.17.4

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 (\sqrt[5]{x^{3}} - 12 \sqrt[5]{x^{2}} + 3 \sqrt[5]{x} \cdot 16 + {(- 4)}^{3}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.17.5

Multiply $16$ by $3$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 (\sqrt[5]{x^{3}} - 12 \sqrt[5]{x^{2}} + 48 \sqrt[5]{x} + {(- 4)}^{3}) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.17.6

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 (\sqrt[5]{x^{3}} - 12 \sqrt[5]{x^{2}} + 48 \sqrt[5]{x} - 64) + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.18

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} + 160 (- 12 \sqrt[5]{x^{2}}) + 160 (48 \sqrt[5]{x}) + 160 \cdot - 64 + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.19

Simplify.

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Step 5.2.3.19.1

Multiply $- 12$ by $160$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 160 (48 \sqrt[5]{x}) + 160 \cdot - 64 + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.19.2

Multiply $48$ by $160$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} + 160 \cdot - 64 + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.19.3

Multiply $160$ by $- 64$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 40960 {(\frac{\sqrt[5]{x} - 4}{8})}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.20

Apply the product rule to $\frac{\sqrt[5]{x} - 4}{8}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 40960 (\frac{{(\sqrt[5]{x} - 4)}^{2}}{8^{2}}) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.21

Raise $8$ to the power of $2$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 40960 (\frac{{(\sqrt[5]{x} - 4)}^{2}}{64}) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.22

Cancel the common factor of $64$ .

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Step 5.2.3.22.1

Factor $64$ out of $40960$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 64 (640) (\frac{{(\sqrt[5]{x} - 4)}^{2}}{64}) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.22.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 64 \cdot (640 (\frac{{(\sqrt[5]{x} - 4)}^{2}}{64})) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.22.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 {(\sqrt[5]{x} - 4)}^{2} + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.23

Rewrite ${(\sqrt[5]{x} - 4)}^{2}$ as $(\sqrt[5]{x} - 4) (\sqrt[5]{x} - 4)$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 ((\sqrt[5]{x} - 4) (\sqrt[5]{x} - 4)) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.24

Expand $(\sqrt[5]{x} - 4) (\sqrt[5]{x} - 4)$ using the FOIL Method.

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Step 5.2.3.24.1

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x} (\sqrt[5]{x} - 4) - 4 (\sqrt[5]{x} - 4)) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.24.2

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot - 4 - 4 (\sqrt[5]{x} - 4)) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.24.3

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot - 4 - 4 \sqrt[5]{x} - 4 \cdot - 4) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.25

Simplify and combine like terms.

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Step 5.2.3.25.1

Simplify each term.

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Step 5.2.3.25.1.1

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

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Step 5.2.3.25.1.1.1

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot - 4 - 4 \sqrt[5]{x} - 4 \cdot - 4) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.25.1.1.2

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot - 4 - 4 \sqrt[5]{x} - 4 \cdot - 4) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.25.1.1.3

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 ({\sqrt[5]{x}}^{1 + 1} + \sqrt[5]{x} \cdot - 4 - 4 \sqrt[5]{x} - 4 \cdot - 4) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.25.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 ({\sqrt[5]{x}}^{2} + \sqrt[5]{x} \cdot - 4 - 4 \sqrt[5]{x} - 4 \cdot - 4) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.25.1.2

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

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x^{2}} + \sqrt[5]{x} \cdot - 4 - 4 \sqrt[5]{x} - 4 \cdot - 4) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.25.1.3

Move $- 4$ to the left of $\sqrt[5]{x}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x^{2}} - 4 \cdot \sqrt[5]{x} - 4 \sqrt[5]{x} - 4 \cdot - 4) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.25.1.4

Multiply $- 4$ by $- 4$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x^{2}} - 4 \sqrt[5]{x} - 4 \sqrt[5]{x} + 16) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.25.2

Subtract $4 \sqrt[5]{x}$ from $- 4 \sqrt[5]{x}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 (\sqrt[5]{x^{2}} - 8 \sqrt[5]{x} + 16) + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.26

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} + 640 (- 8 \sqrt[5]{x}) + 640 \cdot 16 + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.27

Simplify.

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Step 5.2.3.27.1

Multiply $- 8$ by $640$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 640 \cdot 16 + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.27.2

Multiply $640$ by $16$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 10240 (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.28

Cancel the common factor of $8$ .

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Step 5.2.3.28.1

Factor $8$ out of $10240$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 8 (1280) (\frac{\sqrt[5]{x} - 4}{8}) + 1024$

Step 5.2.3.28.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 8 \cdot (1280 (\frac{\sqrt[5]{x} - 4}{8})) + 1024$

Step 5.2.3.28.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 1280 (\sqrt[5]{x} - 4) + 1024$

Step 5.2.3.29

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} + 1280 \cdot - 4 + 1024$

Step 5.2.3.30

Multiply $1280$ by $- 4$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} - 5120 + 1024$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x - 20 \sqrt[5]{x^{4}} + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 + 20 \sqrt[5]{x^{4}} - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} - 5120 + 1024$ .

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Step 5.2.4.1.1

Add $- 20 \sqrt[5]{x^{4}}$ and $20 \sqrt[5]{x^{4}}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 0 + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} - 5120 + 1024$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} + 1920 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} - 1920 \sqrt[5]{x^{2}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} - 5120 + 1024$

Step 5.2.4.1.3

Subtract $1920 \sqrt[5]{x^{2}}$ from $1920 \sqrt[5]{x^{2}}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} + 0 - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} - 5120 + 1024$

Step 5.2.4.1.4

Add $x + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}}$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} - 640 \sqrt[5]{x^{2}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 10240 + 640 \sqrt[5]{x^{2}} - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} - 5120 + 1024$

Step 5.2.4.1.5

Add $- 640 \sqrt[5]{x^{2}}$ and $640 \sqrt[5]{x^{2}}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} + 0 + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 10240 - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} - 5120 + 1024$

Step 5.2.4.1.6

Add $x + 160 \sqrt[5]{x^{3}}$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 10240 - 5120 \sqrt[5]{x} + 10240 + 1280 \sqrt[5]{x} - 5120 + 1024$

Step 5.2.4.1.7

Add $- 10240$ and $10240$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} + 0 - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x} - 5120 + 1024$

Step 5.2.4.1.8

Add $x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 5120 + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x}$ and $0$ .

Step 5.2.4.1.9

Subtract $5120$ from $5120$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 0 + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x} + 1024$

Step 5.2.4.1.10

Add $x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 1024 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x} + 1024$

Step 5.2.4.1.11

Add $- 1024$ and $1024$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} + 0 - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x}$

Step 5.2.4.1.12

Add $x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 320 \sqrt[5]{x^{3}} - 5120 \sqrt[5]{x} + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x}$

Step 5.2.4.2

Subtract $320 \sqrt[5]{x^{3}}$ from $160 \sqrt[5]{x^{3}}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x}$

Step 5.2.4.3

Combine the opposite terms in $x - 160 \sqrt[5]{x^{3}} + 1280 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 160 \sqrt[5]{x^{3}} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x}$ .

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Step 5.2.4.3.1

Add $- 160 \sqrt[5]{x^{3}}$ and $160 \sqrt[5]{x^{3}}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 0 + 1280 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x}$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 1280 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x}$

Step 5.2.4.4

Subtract $5120 \sqrt[5]{x}$ from $1280 \sqrt[5]{x}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 3840 \sqrt[5]{x} + 7680 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x}$

Step 5.2.4.5

Add $- 3840 \sqrt[5]{x}$ and $7680 \sqrt[5]{x}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 3840 \sqrt[5]{x} - 5120 \sqrt[5]{x} + 1280 \sqrt[5]{x}$

Step 5.2.4.6

Subtract $5120 \sqrt[5]{x}$ from $3840 \sqrt[5]{x}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x - 1280 \sqrt[5]{x} + 1280 \sqrt[5]{x}$

Step 5.2.4.7

Combine the opposite terms in $x - 1280 \sqrt[5]{x} + 1280 \sqrt[5]{x}$ .

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Step 5.2.4.7.1

Add $- 1280 \sqrt[5]{x}$ and $1280 \sqrt[5]{x}$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x + 0$

Step 5.2.4.7.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{x} - 4}{8}) = x$

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 $f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024} - 4}{8}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $1024$ out of $32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024$ .

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Step 5.3.3.1.1

Factor $1024$ out of $32768 x^{5}$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5}) + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024} - 4}{8}$

Step 5.3.3.1.2

Factor $1024$ out of $81920 x^{4}$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5}) + 1024 (80 x^{4}) + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024} - 4}{8}$

Step 5.3.3.1.3

Factor $1024$ out of $81920 x^{3}$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5}) + 1024 (80 x^{4}) + 1024 (80 x^{3}) + 40960 x^{2} + 10240 x + 1024} - 4}{8}$

Step 5.3.3.1.4

Factor $1024$ out of $40960 x^{2}$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5}) + 1024 (80 x^{4}) + 1024 (80 x^{3}) + 1024 (40 x^{2}) + 10240 x + 1024} - 4}{8}$

Step 5.3.3.1.5

Factor $1024$ out of $10240 x$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5}) + 1024 (80 x^{4}) + 1024 (80 x^{3}) + 1024 (40 x^{2}) + 1024 (10 x) + 1024} - 4}{8}$

Step 5.3.3.1.6

Factor $1024$ out of $1024$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5}) + 1024 (80 x^{4}) + 1024 (80 x^{3}) + 1024 (40 x^{2}) + 1024 (10 x) + 1024 (1)} - 4}{8}$

Step 5.3.3.1.7

Factor $1024$ out of $1024 (32 x^{5}) + 1024 (80 x^{4})$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5} + 80 x^{4}) + 1024 (80 x^{3}) + 1024 (40 x^{2}) + 1024 (10 x) + 1024 (1)} - 4}{8}$

Step 5.3.3.1.8

Factor $1024$ out of $1024 (32 x^{5} + 80 x^{4}) + 1024 (80 x^{3})$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5} + 80 x^{4} + 80 x^{3}) + 1024 (40 x^{2}) + 1024 (10 x) + 1024 (1)} - 4}{8}$

Step 5.3.3.1.9

Factor $1024$ out of $1024 (32 x^{5} + 80 x^{4} + 80 x^{3}) + 1024 (40 x^{2})$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5} + 80 x^{4} + 80 x^{3} + 40 x^{2}) + 1024 (10 x) + 1024 (1)} - 4}{8}$

Step 5.3.3.1.10

Factor $1024$ out of $1024 (32 x^{5} + 80 x^{4} + 80 x^{3} + 40 x^{2}) + 1024 (10 x)$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5} + 80 x^{4} + 80 x^{3} + 40 x^{2} + 10 x) + 1024 (1)} - 4}{8}$

Step 5.3.3.1.11

Factor $1024$ out of $1024 (32 x^{5} + 80 x^{4} + 80 x^{3} + 40 x^{2} + 10 x) + 1024 (1)$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 (32 x^{5} + 80 x^{4} + 80 x^{3} + 40 x^{2} + 10 x + 1)} - 4}{8}$

Step 5.3.3.2

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

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 ({(2 x)}^{5} + 5 {(2 x)}^{4} + 10 {(2 x)}^{3} + 10 {(2 x)}^{2} + 5 (2 x) + 1)} - 4}{8}$

Step 5.3.3.3

Factor ${(2 x)}^{5} + 5 {(2 x)}^{4} + 10 {(2 x)}^{3} + 10 {(2 x)}^{2} + 5 (2 x) + 1$ using the binomial theorem.

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{1024 {(2 x + 1)}^{5}} - 4}{8}$

Step 5.3.3.4

Rewrite $1024 {(2 x + 1)}^{5}$ as ${(4 (2 x + 1))}^{5}$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{\sqrt[5]{{(4 (2 x + 1))}^{5}} - 4}{8}$

Step 5.3.3.5

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

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{4 (2 x + 1) - 4}{8}$

Step 5.3.3.6

Apply the distributive property.

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{4 (2 x) + 4 \cdot 1 - 4}{8}$

Step 5.3.3.7

Multiply $2$ by $4$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{8 x + 4 \cdot 1 - 4}{8}$

Step 5.3.3.8

Multiply $4$ by $1$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{8 x + 4 - 4}{8}$

Step 5.3.3.9

Subtract $4$ from $4$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{8 x + 0}{8}$

Step 5.3.3.10

Add $8 x$ and $0$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{8 x}{8}$

Step 5.3.4

Cancel the common factor of $8$ .

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Step 5.3.4.1

Cancel the common factor.

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = \frac{8 x}{8}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024$ is the inverse of $f (x) = \frac{\sqrt[5]{x} - 4}{8}$ .

$f^{- 1} (x) = 32768 x^{5} + 81920 x^{4} + 81920 x^{3} + 40960 x^{2} + 10240 x + 1024$