Algebra Examples

$y = 9 \log (- 4 x^{5} + 8) - 1$

Step 1

Interchange the variables.

$x = 9 \log (- 4 y^{5} + 8) - 1$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $9 \log (- 4 y^{5} + 8) - 1 = x$ .

$9 \log (- 4 y^{5} + 8) - 1 = x$

Step 2.2

Simplify $9 \log (- 4 y^{5} + 8)$ by moving $9$ inside the logarithm.

$\log ({(- 4 y^{5} + 8)}^{9}) - 1 = x$

Step 2.3

Add $1$ to both sides of the equation.

$\log ({(- 4 y^{5} + 8)}^{9}) = x + 1$

Step 2.4

Rewrite $\log ({(- 4 y^{5} + 8)}^{9}) = x + 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{x + 1} = {(- 4 y^{5} + 8)}^{9}$

Step 2.5

Solve for $y$ .

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Step 2.5.1

Rewrite the equation as ${(- 4 y^{5} + 8)}^{9} = 10^{x + 1}$ .

${(- 4 y^{5} + 8)}^{9} = 10^{x + 1}$

Step 2.5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$- 4 y^{5} + 8 = \sqrt[9]{10^{x + 1}}$

Step 2.5.3

Subtract $8$ from both sides of the equation.

$- 4 y^{5} = \sqrt[9]{10^{x + 1}} - 8$

Step 2.5.4

Divide each term in $- 4 y^{5} = \sqrt[9]{10^{x + 1}} - 8$ by $- 4$ and simplify.

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Step 2.5.4.1

Divide each term in $- 4 y^{5} = \sqrt[9]{10^{x + 1}} - 8$ by $- 4$ .

$\frac{- 4 y^{5}}{- 4} = \frac{\sqrt[9]{10^{x + 1}}}{- 4} + \frac{- 8}{- 4}$

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $- 4$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

$\frac{- 4 y^{5}}{- 4} = \frac{\sqrt[9]{10^{x + 1}}}{- 4} + \frac{- 8}{- 4}$

Step 2.5.4.2.1.2

Divide $y^{5}$ by $1$ .

$y^{5} = \frac{\sqrt[9]{10^{x + 1}}}{- 4} + \frac{- 8}{- 4}$

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Simplify each term.

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Step 2.5.4.3.1.1

Move the negative in front of the fraction.

$y^{5} = - \frac{\sqrt[9]{10^{x + 1}}}{4} + \frac{- 8}{- 4}$

Step 2.5.4.3.1.2

Divide $- 8$ by $- 4$ .

$y^{5} = - \frac{\sqrt[9]{10^{x + 1}}}{4} + 2$

Step 2.5.4.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y^{5} = - \frac{\sqrt[9]{10^{x + 1}}}{4} + 2 \cdot \frac{4}{4}$

Step 2.5.4.3.3

Combine $2$ and $\frac{4}{4}$ .

$y^{5} = - \frac{\sqrt[9]{10^{x + 1}}}{4} + \frac{2 \cdot 4}{4}$

Step 2.5.4.3.4

Combine the numerators over the common denominator.

$y^{5} = \frac{- \sqrt[9]{10^{x + 1}} + 2 \cdot 4}{4}$

Step 2.5.4.3.5

Multiply $2$ by $4$ .

$y^{5} = \frac{- \sqrt[9]{10^{x + 1}} + 8}{4}$

Step 2.5.4.3.6

Factor $- 1$ out of $- \sqrt[9]{10^{x + 1}}$ .

$y^{5} = \frac{- (\sqrt[9]{10^{x + 1}}) + 8}{4}$

Step 2.5.4.3.7

Rewrite $8$ as $- 1 (- 8)$ .

$y^{5} = \frac{- (\sqrt[9]{10^{x + 1}}) - 1 (- 8)}{4}$

Step 2.5.4.3.8

Factor $- 1$ out of $- (\sqrt[9]{10^{x + 1}}) - 1 (- 8)$ .

$y^{5} = \frac{- (\sqrt[9]{10^{x + 1}} - 8)}{4}$

Step 2.5.4.3.9

Simplify the expression.

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Step 2.5.4.3.9.1

Rewrite $- (\sqrt[9]{10^{x + 1}} - 8)$ as $- 1 (\sqrt[9]{10^{x + 1}} - 8)$ .

$y^{5} = \frac{- 1 (\sqrt[9]{10^{x + 1}} - 8)}{4}$

Step 2.5.4.3.9.2

Move the negative in front of the fraction.

$y^{5} = - \frac{\sqrt[9]{10^{x + 1}} - 8}{4}$

Step 2.5.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[5]{- \frac{\sqrt[9]{10^{x + 1}} - 8}{4}}$

Step 2.5.6

Simplify $\sqrt[5]{- \frac{\sqrt[9]{10^{x + 1}} - 8}{4}}$ .

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Step 2.5.6.1

Rewrite $- \frac{\sqrt[9]{10^{x + 1}} - 8}{4}$ as ${({(- 1)}^{5})}^{5} \frac{\sqrt[9]{10^{x + 1}} - 8}{4}$ .

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Step 2.5.6.1.1

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$y = \sqrt[5]{{(- 1)}^{5} \frac{\sqrt[9]{10^{x + 1}} - 8}{4}}$

Step 2.5.6.1.2

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$y = \sqrt[5]{{({(- 1)}^{5})}^{5} \frac{\sqrt[9]{10^{x + 1}} - 8}{4}}$

Step 2.5.6.2

Pull terms out from under the radical.

$y = {(- 1)}^{5} \sqrt[5]{\frac{\sqrt[9]{10^{x + 1}} - 8}{4}}$

Step 2.5.6.3

Raise $- 1$ to the power of $5$ .

$y = - \sqrt[5]{\frac{\sqrt[9]{10^{x + 1}} - 8}{4}}$

Step 2.5.6.4

Rewrite $\sqrt[5]{\frac{\sqrt[9]{10^{x + 1}} - 8}{4}}$ as $\frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8}}{\sqrt[5]{4}}$ .

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8}}{\sqrt[5]{4}}$

Step 2.5.6.5

Multiply $\frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8}}{\sqrt[5]{4}}$ by $\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}}$ .

$y = - (\frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8}}{\sqrt[5]{4}} \cdot \frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}})$

Step 2.5.6.6

Combine and simplify the denominator.

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Step 2.5.6.6.1

Multiply $\frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8}}{\sqrt[5]{4}}$ by $\frac{{\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{4}}$ .

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{\sqrt[5]{4} {\sqrt[5]{4}}^{4}}$

Step 2.5.6.6.2

Raise $\sqrt[5]{4}$ to the power of $1$ .

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{1} {\sqrt[5]{4}}^{4}}$

Step 2.5.6.6.3

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

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{1 + 4}}$

Step 2.5.6.6.4

Add $1$ and $4$ .

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{{\sqrt[5]{4}}^{5}}$

Step 2.5.6.6.5

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

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Step 2.5.6.6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4}$ as $4^{\frac{1}{5}}$ .

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{{(4^{\frac{1}{5}})}^{5}}$

Step 2.5.6.6.5.2

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

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{4^{\frac{1}{5} \cdot 5}}$

Step 2.5.6.6.5.3

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

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{4^{\frac{5}{5}}}$

Step 2.5.6.6.5.4

Cancel the common factor of $5$ .

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Step 2.5.6.6.5.4.1

Cancel the common factor.

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{4^{\frac{5}{5}}}$

Step 2.5.6.6.5.4.2

Rewrite the expression.

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{4^{1}}$

Step 2.5.6.6.5.5

Evaluate the exponent.

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} {\sqrt[5]{4}}^{4}}{4}$

Step 2.5.6.7

Simplify the numerator.

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Step 2.5.6.7.1

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

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} \sqrt[5]{4^{4}}}{4}$

Step 2.5.6.7.2

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

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} \sqrt[5]{256}}{4}$

Step 2.5.6.7.3

Rewrite $256$ as $2^{5} \cdot 8$ .

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Step 2.5.6.7.3.1

Factor $32$ out of $256$ .

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} \sqrt[5]{32 (8)}}{4}$

Step 2.5.6.7.3.2

Rewrite $32$ as $2^{5}$ .

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} \sqrt[5]{2^{5} \cdot 8}}{4}$

Step 2.5.6.7.4

Pull terms out from under the radical.

$y = - \frac{\sqrt[5]{\sqrt[9]{10^{x + 1}} - 8} \cdot 2 \sqrt[5]{8}}{4}$

Step 2.5.6.7.5

Combine using the product rule for radicals.

$y = - \frac{2 \sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{4}$

Step 2.5.6.8

Cancel the common factor of $2$ and $4$ .

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Step 2.5.6.8.1

Factor $2$ out of $2 \sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}$ .

$y = - \frac{2 (\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8})}{4}$

Step 2.5.6.8.2

Cancel the common factors.

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Step 2.5.6.8.2.1

Factor $2$ out of $4$ .

$y = - \frac{2 \sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2 \cdot 2}$

Step 2.5.6.8.2.2

Cancel the common factor.

$y = - \frac{2 \sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2 \cdot 2}$

Step 2.5.6.8.2.3

Rewrite the expression.

$y = - \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}$

Step 3

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

$f^{- 1} (x) = - \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}$

Step 4

Verify if $f^{- 1} (x) = - \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}$ is the inverse of $f (x) = 9 \log (- 4 x^{5} + 8) - 1$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (9 \log (- 4 x^{5} + 8) - 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(\sqrt[9]{10^{(9 \log (- 4 x^{5} + 8) - 1) + 1}} - 8) \cdot 8}}{2}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{10^{9 \log (- 4 x^{5} + 8) - 1 + 1}}$ as $10^{\frac{9 \log (- 4 x^{5} + 8) - 1 + 1}{9}}$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(10^{\frac{9 \log (- 4 x^{5} + 8) - 1 + 1}{9}} - 8) \cdot 8}}{2}$

Step 4.2.3.2

Simplify the numerator.

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Step 4.2.3.2.1

Simplify $9 \log (- 4 x^{5} + 8)$ by moving $9$ inside the logarithm.

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(10^{\frac{\log ({(- 4 x^{5} + 8)}^{9}) - 1 + 1}{9}} - 8) \cdot 8}}{2}$

Step 4.2.3.2.2

Add $- 1$ and $1$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(10^{\frac{\log ({(- 4 x^{5} + 8)}^{9}) + 0}{9}} - 8) \cdot 8}}{2}$

Step 4.2.3.2.3

Add $\log ({(- 4 x^{5} + 8)}^{9})$ and $0$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(10^{\frac{\log ({(- 4 x^{5} + 8)}^{9})}{9}} - 8) \cdot 8}}{2}$

Step 4.2.3.3

Expand $\log ({(- 4 x^{5} + 8)}^{9})$ by moving $9$ outside the logarithm.

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(10^{\frac{9 \log (- 4 x^{5} + 8)}{9}} - 8) \cdot 8}}{2}$

Step 4.2.3.4

Cancel the common factor of $9$ .

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Step 4.2.3.4.1

Cancel the common factor.

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(10^{\frac{9 \log (- 4 x^{5} + 8)}{9}} - 8) \cdot 8}}{2}$

Step 4.2.3.4.2

Divide $\log (- 4 x^{5} + 8)$ by $1$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(10^{\log (- 4 x^{5} + 8)} - 8) \cdot 8}}{2}$

Step 4.2.3.5

Exponentiation and log are inverse functions.

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(- 4 x^{5} + 8 - 8) \cdot 8}}{2}$

Step 4.2.3.6

Subtract $8$ from $8$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{(- 4 x^{5} + 0) \cdot 8}}{2}$

Step 4.2.3.7

Add $- 4 x^{5}$ and $0$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{- 4 x^{5} \cdot 8}}{2}$

Step 4.2.3.8

Multiply $8$ by $- 4$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{- 32 x^{5}}}{2}$

Step 4.2.3.9

Rewrite $- 32 x^{5}$ as ${(- 2 x)}^{5}$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{\sqrt[5]{{(- 2 x)}^{5}}}{2}$

Step 4.2.3.10

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

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{- 2 x}{2}$

Step 4.2.4

Cancel the common factor of $- 2$ and $2$ .

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Step 4.2.4.1

Factor $2$ out of $- 2 x$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{2 (- x)}{2}$

Step 4.2.4.2

Cancel the common factors.

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Step 4.2.4.2.1

Factor $2$ out of $2$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{2 (- x)}{2 (1)}$

Step 4.2.4.2.2

Cancel the common factor.

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{2 (- x)}{2 \cdot 1}$

Step 4.2.4.2.3

Rewrite the expression.

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = - \frac{- x}{1}$

Step 4.2.4.2.4

Divide $- x$ by $1$ .

$f^{- 1} (9 \log (- 4 x^{5} + 8) - 1) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 {(- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2})}^{5} + 8) - 1$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.3.1.1.1

Apply the product rule to $- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 ({(- 1)}^{5} {(\frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2})}^{5}) + 8) - 1$

Step 4.3.3.1.1.2

Apply the product rule to $\frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 ({(- 1)}^{5} (\frac{{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}^{5}}{2^{5}})) + 8) - 1$

Step 4.3.3.1.2

Raise $- 1$ to the power of $5$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}^{5}}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3

Simplify the numerator.

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Step 4.3.3.1.3.1

Rewrite ${\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}^{5}$ as $(\sqrt[9]{10^{x + 1}} - 8) \cdot 8$ .

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Step 4.3.3.1.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}$ as ${((\sqrt[9]{10^{x + 1}} - 8) \cdot 8)}^{\frac{1}{5}}$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{{({((\sqrt[9]{10^{x + 1}} - 8) \cdot 8)}^{\frac{1}{5}})}^{5}}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.1.2

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

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{{((\sqrt[9]{10^{x + 1}} - 8) \cdot 8)}^{\frac{1}{5} \cdot 5}}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.1.3

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

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{{((\sqrt[9]{10^{x + 1}} - 8) \cdot 8)}^{\frac{5}{5}}}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.1.4

Cancel the common factor of $5$ .

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Step 4.3.3.1.3.1.4.1

Cancel the common factor.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{{((\sqrt[9]{10^{x + 1}} - 8) \cdot 8)}^{\frac{5}{5}}}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.1.4.2

Rewrite the expression.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.1.5

Simplify.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.2

Apply the distributive property.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{\sqrt[9]{10^{x + 1}} \cdot 8 - 8 \cdot 8}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.3

Move $8$ to the left of $\sqrt[9]{10^{x + 1}}$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{8 \cdot \sqrt[9]{10^{x + 1}} - 8 \cdot 8}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.4

Multiply $- 8$ by $8$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{8 \cdot \sqrt[9]{10^{x + 1}} - 64}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.5

Factor $8$ out of $8 \sqrt[9]{10^{x + 1}} - 64$ .

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Step 4.3.3.1.3.5.1

Factor $8$ out of $8 \sqrt[9]{10^{x + 1}}$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{8 (\sqrt[9]{10^{x + 1}}) - 64}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.5.2

Factor $8$ out of $- 64$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{8 \sqrt[9]{10^{x + 1}} + 8 \cdot - 8}{2^{5}}) + 8) - 1$

Step 4.3.3.1.3.5.3

Factor $8$ out of $8 \sqrt[9]{10^{x + 1}} + 8 \cdot - 8$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{8 (\sqrt[9]{10^{x + 1}} - 8)}{2^{5}}) + 8) - 1$

Step 4.3.3.1.4

Raise $2$ to the power of $5$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 (- \frac{8 (\sqrt[9]{10^{x + 1}} - 8)}{32}) + 8) - 1$

Step 4.3.3.1.5

Cancel the common factor of $4$ .

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Step 4.3.3.1.5.1

Move the leading negative in $- \frac{8 (\sqrt[9]{10^{x + 1}} - 8)}{32}$ into the numerator.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 4 \frac{- 8 (\sqrt[9]{10^{x + 1}} - 8)}{32} + 8) - 1$

Step 4.3.3.1.5.2

Factor $4$ out of $- 4$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (4 (- 1) (\frac{- 8 (\sqrt[9]{10^{x + 1}} - 8)}{32}) + 8) - 1$

Step 4.3.3.1.5.3

Factor $4$ out of $32$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (4 \cdot (- 1 \frac{- 8 (\sqrt[9]{10^{x + 1}} - 8)}{4 \cdot 8}) + 8) - 1$

Step 4.3.3.1.5.4

Cancel the common factor.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (4 \cdot (- 1 \frac{- 8 (\sqrt[9]{10^{x + 1}} - 8)}{4 \cdot 8}) + 8) - 1$

Step 4.3.3.1.5.5

Rewrite the expression.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 \frac{- 8 (\sqrt[9]{10^{x + 1}} - 8)}{8} + 8) - 1$

Step 4.3.3.1.6

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

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Step 4.3.3.1.6.1

Factor $8$ out of $- 8 (\sqrt[9]{10^{x + 1}} - 8)$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 \frac{8 (- (\sqrt[9]{10^{x + 1}} - 8))}{8} + 8) - 1$

Step 4.3.3.1.6.2

Cancel the common factors.

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Step 4.3.3.1.6.2.1

Factor $8$ out of $8$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 \frac{8 (- (\sqrt[9]{10^{x + 1}} - 8))}{8 (1)} + 8) - 1$

Step 4.3.3.1.6.2.2

Cancel the common factor.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 \frac{8 (- (\sqrt[9]{10^{x + 1}} - 8))}{8 \cdot 1} + 8) - 1$

Step 4.3.3.1.6.2.3

Rewrite the expression.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 \frac{- (\sqrt[9]{10^{x + 1}} - 8)}{1} + 8) - 1$

Step 4.3.3.1.6.2.4

Divide $- (\sqrt[9]{10^{x + 1}} - 8)$ by $1$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 (- (\sqrt[9]{10^{x + 1}} - 8)) + 8) - 1$

Step 4.3.3.1.7

Apply the distributive property.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 (- \sqrt[9]{10^{x + 1}} + 8) + 8) - 1$

Step 4.3.3.1.8

Multiply $- 1$ by $- 8$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 (- \sqrt[9]{10^{x + 1}} + 8) + 8) - 1$

Step 4.3.3.1.9

Apply the distributive property.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (- 1 (- \sqrt[9]{10^{x + 1}}) - 1 \cdot 8 + 8) - 1$

Step 4.3.3.1.10

Multiply $- 1 (- \sqrt[9]{10^{x + 1}})$ .

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Step 4.3.3.1.10.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (1 \sqrt[9]{10^{x + 1}} - 1 \cdot 8 + 8) - 1$

Step 4.3.3.1.10.2

Multiply $\sqrt[9]{10^{x + 1}}$ by $1$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (\sqrt[9]{10^{x + 1}} - 1 \cdot 8 + 8) - 1$

Step 4.3.3.1.11

Multiply $- 1$ by $8$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (\sqrt[9]{10^{x + 1}} - 8 + 8) - 1$

Step 4.3.3.2

Combine the opposite terms in $\sqrt[9]{10^{x + 1}} - 8 + 8$ .

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Step 4.3.3.2.1

Add $- 8$ and $8$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (\sqrt[9]{10^{x + 1}} + 0) - 1$

Step 4.3.3.2.2

Add $\sqrt[9]{10^{x + 1}}$ and $0$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = 9 \log (\sqrt[9]{10^{x + 1}}) - 1$

Step 4.3.3.3

Simplify $9 \log (\sqrt[9]{10^{x + 1}})$ by moving $9$ inside the logarithm.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = \log ({\sqrt[9]{10^{x + 1}}}^{9}) - 1$

Step 4.3.3.4

Rewrite ${\sqrt[9]{10^{x + 1}}}^{9}$ as $10^{x + 1}$ .

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Step 4.3.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{10^{x + 1}}$ as $10^{\frac{x + 1}{9}}$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = \log ({(10^{\frac{x + 1}{9}})}^{9}) - 1$

Step 4.3.3.4.2

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

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = \log (10^{\frac{x + 1}{9} \cdot 9}) - 1$

Step 4.3.3.4.3

Combine $\frac{x + 1}{9}$ and $9$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = \log (10^{\frac{(x + 1) \cdot 9}{9}}) - 1$

Step 4.3.3.4.4

Cancel the common factor of $9$ .

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Step 4.3.3.4.4.1

Cancel the common factor.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = \log (10^{\frac{(x + 1) \cdot 9}{9}}) - 1$

Step 4.3.3.4.4.2

Divide $x + 1$ by $1$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = \log (10^{x + 1}) - 1$

Step 4.3.3.5

Use logarithm rules to move $x + 1$ out of the exponent.

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = (x + 1) \log (10) - 1$

Step 4.3.3.6

Logarithm base $10$ of $10$ is $1$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = (x + 1) \cdot 1 - 1$

Step 4.3.3.7

Multiply $x + 1$ by $1$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = x + 1 - 1$

Step 4.3.4

Combine the opposite terms in $x + 1 - 1$ .

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Step 4.3.4.1

Subtract $1$ from $1$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (- \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}$ is the inverse of $f (x) = 9 \log (- 4 x^{5} + 8) - 1$ .

$f^{- 1} (x) = - \frac{\sqrt[5]{(\sqrt[9]{10^{x + 1}} - 8) \cdot 8}}{2}$