Algebra Examples

$f (x) = \frac{\sqrt[5]{3 (x + 8)}}{10}$

Step 1

Write $f (x) = \frac{\sqrt[5]{3 (x + 8)}}{10}$ as an equation.

$y = \frac{\sqrt[5]{3 (x + 8)}}{10}$

Step 2

Interchange the variables.

$x = \frac{\sqrt[5]{3 (y + 8)}}{10}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{\sqrt[5]{3 (y + 8)}}{10} = x$ .

$\frac{\sqrt[5]{3 (y + 8)}}{10} = x$

Step 3.2

Multiply both sides by $10$ .

$\frac{\sqrt[5]{3 (y + 8)}}{10} \cdot 10 = x \cdot 10$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $10$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{\sqrt[5]{3 (y + 8)}}{10} \cdot 10 = x \cdot 10$

Step 3.3.1.1.2

Rewrite the expression.

$\sqrt[5]{3 (y + 8)} = x \cdot 10$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Move $10$ to the left of $x$ .

$\sqrt[5]{3 (y + 8)} = 10 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

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

${\sqrt[5]{3 (y + 8)}}^{5} = {(10 x)}^{5}$

Step 3.4.2

Simplify each side of the equation.

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Step 3.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3 (y + 8)}$ as ${(3 (y + 8))}^{\frac{1}{5}}$ .

${({(3 (y + 8))}^{\frac{1}{5}})}^{5} = {(10 x)}^{5}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Simplify ${({(3 (y + 8))}^{\frac{1}{5}})}^{5}$ .

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Step 3.4.2.2.1.1

Multiply the exponents in ${({(3 (y + 8))}^{\frac{1}{5}})}^{5}$ .

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Step 3.4.2.2.1.1.1

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

${(3 (y + 8))}^{\frac{1}{5} \cdot 5} = {(10 x)}^{5}$

Step 3.4.2.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.4.2.2.1.1.2.1

Cancel the common factor.

${(3 (y + 8))}^{\frac{1}{5} \cdot 5} = {(10 x)}^{5}$

Step 3.4.2.2.1.1.2.2

Rewrite the expression.

${(3 (y + 8))}^{1} = {(10 x)}^{5}$

Step 3.4.2.2.1.2

Apply the distributive property.

${(3 y + 3 \cdot 8)}^{1} = {(10 x)}^{5}$

Step 3.4.2.2.1.3

Multiply.

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Step 3.4.2.2.1.3.1

Multiply $3$ by $8$ .

${(3 y + 24)}^{1} = {(10 x)}^{5}$

Step 3.4.2.2.1.3.2

Simplify.

$3 y + 24 = {(10 x)}^{5}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify ${(10 x)}^{5}$ .

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Step 3.4.2.3.1.1

Apply the product rule to $10 x$ .

$3 y + 24 = 10^{5} x^{5}$

Step 3.4.2.3.1.2

Raise $10$ to the power of $5$ .

$3 y + 24 = 100000 x^{5}$

Step 3.4.3

Solve for $y$ .

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Step 3.4.3.1

Subtract $24$ from both sides of the equation.

$3 y = 100000 x^{5} - 24$

Step 3.4.3.2

Divide each term in $3 y = 100000 x^{5} - 24$ by $3$ and simplify.

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Step 3.4.3.2.1

Divide each term in $3 y = 100000 x^{5} - 24$ by $3$ .

$\frac{3 y}{3} = \frac{100000 x^{5}}{3} + \frac{- 24}{3}$

Step 3.4.3.2.2

Simplify the left side.

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Step 3.4.3.2.2.1

Cancel the common factor of $3$ .

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Step 3.4.3.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{100000 x^{5}}{3} + \frac{- 24}{3}$

Step 3.4.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{100000 x^{5}}{3} + \frac{- 24}{3}$

Step 3.4.3.2.3

Simplify the right side.

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Step 3.4.3.2.3.1

Divide $- 24$ by $3$ .

$y = \frac{100000 x^{5}}{3} - 8$

Step 4

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

$f^{- 1} (x) = \frac{100000 x^{5}}{3} - 8$

Step 5

Verify if $f^{- 1} (x) = \frac{100000 x^{5}}{3} - 8$ is the inverse of $f (x) = \frac{\sqrt[5]{3 (x + 8)}}{10}$ .

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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 $f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 {(\frac{\sqrt[5]{3 (x + 8)}}{10})}^{5}}{3} - 8$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Apply the product rule to $\frac{\sqrt[5]{3 (x + 8)}}{10}$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{{\sqrt[5]{3 (x + 8)}}^{5}}{10^{5}})}{3} - 8$

Step 5.2.3.1.2

Simplify the numerator.

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Step 5.2.3.1.2.1

Rewrite ${\sqrt[5]{3 (x + 8)}}^{5}$ as $3 (x + 8)$ .

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Step 5.2.3.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3 (x + 8)}$ as ${(3 (x + 8))}^{\frac{1}{5}}$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{{({(3 (x + 8))}^{\frac{1}{5}})}^{5}}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.1.2

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

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{{(3 (x + 8))}^{\frac{1}{5} \cdot 5}}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.1.3

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

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{{(3 (x + 8))}^{\frac{5}{5}}}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.1.4

Cancel the common factor of $5$ .

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Step 5.2.3.1.2.1.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{{(3 (x + 8))}^{\frac{5}{5}}}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.1.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{3 (x + 8)}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.1.5

Simplify.

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{3 (x + 8)}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.2

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{3 x + 3 \cdot 8}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.3

Multiply $3$ by $8$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{3 x + 24}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.4

Factor $3$ out of $3 x + 24$ .

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Step 5.2.3.1.2.4.1

Factor $3$ out of $3 x$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{3 (x) + 24}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.4.2

Factor $3$ out of $24$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{3 x + 3 \cdot 8}{10^{5}})}{3} - 8$

Step 5.2.3.1.2.4.3

Factor $3$ out of $3 x + 3 \cdot 8$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{3 (x + 8)}{10^{5}})}{3} - 8$

Step 5.2.3.1.3

Raise $10$ to the power of $5$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{100000 (\frac{3 (x + 8)}{100000})}{3} - 8$

Step 5.2.3.2

Combine $100000$ and $\frac{3 (x + 8)}{100000}$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{\frac{100000 (3 (x + 8))}{100000}}{3} - 8$

Step 5.2.3.3

Multiply $100000$ by $3$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{\frac{300000 (x + 8)}{100000}}{3} - 8$

Step 5.2.3.4

Reduce the expression by cancelling the common factors.

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Step 5.2.3.4.1

Reduce the expression $\frac{300000 (x + 8)}{100000}$ by cancelling the common factors.

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Step 5.2.3.4.1.1

Factor $100000$ out of $300000 (x + 8)$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{\frac{100000 (3 (x + 8))}{100000}}{3} - 8$

Step 5.2.3.4.1.2

Factor $100000$ out of $100000$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{\frac{100000 (3 (x + 8))}{100000 (1)}}{3} - 8$

Step 5.2.3.4.1.3

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{\frac{100000 (3 (x + 8))}{100000 \cdot 1}}{3} - 8$

Step 5.2.3.4.1.4

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{\frac{3 (x + 8)}{1}}{3} - 8$

Step 5.2.3.4.2

Divide $3 (x + 8)$ by $1$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{3 (x + 8)}{3} - 8$

Step 5.2.3.5

Cancel the common factor of $3$ .

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Step 5.2.3.5.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = \frac{3 (x + 8)}{3} - 8$

Step 5.2.3.5.2

Divide $x + 8$ by $1$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = x + 8 - 8$

Step 5.2.4

Combine the opposite terms in $x + 8 - 8$ .

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Step 5.2.4.1

Subtract $8$ from $8$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{3 (x + 8)}}{10}) = x$

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 $f (\frac{100000 x^{5}}{3} - 8)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 ((\frac{100000 x^{5}}{3} - 8) + 8)}}{10}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5}}{3} - 8 \cdot \frac{3}{3} + 8)}}{10}$

Step 5.3.3.2

Combine $- 8$ and $\frac{3}{3}$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5}}{3} + \frac{- 8 \cdot 3}{3} + 8)}}{10}$

Step 5.3.3.3

Combine the numerators over the common denominator.

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5} - 8 \cdot 3}{3} + 8)}}{10}$

Step 5.3.3.4

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5} - 8 \cdot 3}{3} + 8 \cdot \frac{3}{3})}}{10}$

Step 5.3.3.5

Combine $8$ and $\frac{3}{3}$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5} - 8 \cdot 3}{3} + \frac{8 \cdot 3}{3})}}{10}$

Step 5.3.3.6

Combine the numerators over the common denominator.

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5} - 8 \cdot 3 + 8 \cdot 3}{3})}}{10}$

Step 5.3.3.7

Reorder terms.

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5} + 3 \cdot 8 - 8 \cdot 3}{3})}}{10}$

Step 5.3.3.8

Rewrite $\frac{100000 x^{5} + 3 \cdot 8 - 8 \cdot 3}{3}$ in a factored form.

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Step 5.3.3.8.1

Multiply $3$ by $8$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5} + 24 - 8 \cdot 3}{3})}}{10}$

Step 5.3.3.8.2

Multiply $- 8$ by $3$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5} + 24 - 24}{3})}}{10}$

Step 5.3.3.8.3

Subtract $24$ from $24$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5} + 0}{3})}}{10}$

Step 5.3.3.8.4

Add $100000 x^{5}$ and $0$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{3 (\frac{100000 x^{5}}{3})}}{10}$

Step 5.3.3.9

Combine $3$ and $\frac{100000 x^{5}}{3}$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{\frac{3 (100000 x^{5})}{3}}}{10}$

Step 5.3.3.10

Multiply $3$ by $100000$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{\frac{300000 x^{5}}{3}}}{10}$

Step 5.3.3.11

Reduce the expression by cancelling the common factors.

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Step 5.3.3.11.1

Reduce the expression $\frac{300000 x^{5}}{3}$ by cancelling the common factors.

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Step 5.3.3.11.1.1

Factor $3$ out of $300000 x^{5}$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{\frac{3 (100000 x^{5})}{3}}}{10}$

Step 5.3.3.11.1.2

Factor $3$ out of $3$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{\frac{3 (100000 x^{5})}{3 (1)}}}{10}$

Step 5.3.3.11.1.3

Cancel the common factor.

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{\frac{3 (100000 x^{5})}{3 \cdot 1}}}{10}$

Step 5.3.3.11.1.4

Rewrite the expression.

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{\frac{100000 x^{5}}{1}}}{10}$

Step 5.3.3.11.2

Divide $100000 x^{5}$ by $1$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{100000 x^{5}}}{10}$

Step 5.3.3.12

Rewrite $100000 x^{5}$ as ${(10 x)}^{5}$ .

$f (\frac{100000 x^{5}}{3} - 8) = \frac{\sqrt[5]{{(10 x)}^{5}}}{10}$

Step 5.3.3.13

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

$f (\frac{100000 x^{5}}{3} - 8) = \frac{10 x}{10}$

Step 5.3.4

Cancel the common factor of $10$ .

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Step 5.3.4.1

Cancel the common factor.

$f (\frac{100000 x^{5}}{3} - 8) = \frac{10 x}{10}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (\frac{100000 x^{5}}{3} - 8) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{100000 x^{5}}{3} - 8$ is the inverse of $f (x) = \frac{\sqrt[5]{3 (x + 8)}}{10}$ .

$f^{- 1} (x) = \frac{100000 x^{5}}{3} - 8$