Algebra Examples

$f (x) = \frac{1}{x^{9}}$ $f^{- 1} (x) = \frac{\sqrt[9]{x^{8}}}{x}$

Step 1

Write $f (x) = \frac{1}{x^{9}}$ as an equation.

$y = \frac{1}{x^{9}}$

Step 2

Interchange the variables.

$x = \frac{1}{y^{9}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{y^{9}} = x$ .

$\frac{1}{y^{9}} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{9}, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$y^{9}$

Step 3.3

Multiply each term in $\frac{1}{y^{9}} = x$ by $y^{9}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{1}{y^{9}} = x$ by $y^{9}$ .

$\frac{1}{y^{9}} y^{9} = x y^{9}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y^{9}$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1}{y^{9}} y^{9} = x y^{9}$

Step 3.3.2.1.2

Rewrite the expression.

$1 = x y^{9}$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $x y^{9} = 1$ .

$x y^{9} = 1$

Step 3.4.2

Divide each term in $x y^{9} = 1$ by $x$ and simplify.

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Step 3.4.2.1

Divide each term in $x y^{9} = 1$ by $x$ .

$\frac{x y^{9}}{x} = \frac{1}{x}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $x$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{x y^{9}}{x} = \frac{1}{x}$

Step 3.4.2.2.1.2

Divide $y^{9}$ by $1$ .

$y^{9} = \frac{1}{x}$

Step 3.4.3

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

$y = \sqrt[9]{\frac{1}{x}}$

Step 3.4.4

Simplify $\sqrt[9]{\frac{1}{x}}$ .

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Step 3.4.4.1

Rewrite $\sqrt[9]{\frac{1}{x}}$ as $\frac{\sqrt[9]{1}}{\sqrt[9]{x}}$ .

$y = \frac{\sqrt[9]{1}}{\sqrt[9]{x}}$

Step 3.4.4.2

Any root of $1$ is $1$ .

$y = \frac{1}{\sqrt[9]{x}}$

Step 3.4.4.3

Multiply $\frac{1}{\sqrt[9]{x}}$ by $\frac{{\sqrt[9]{x}}^{8}}{{\sqrt[9]{x}}^{8}}$ .

$y = \frac{1}{\sqrt[9]{x}} \cdot \frac{{\sqrt[9]{x}}^{8}}{{\sqrt[9]{x}}^{8}}$

Step 3.4.4.4

Combine and simplify the denominator.

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Step 3.4.4.4.1

Multiply $\frac{1}{\sqrt[9]{x}}$ by $\frac{{\sqrt[9]{x}}^{8}}{{\sqrt[9]{x}}^{8}}$ .

$y = \frac{{\sqrt[9]{x}}^{8}}{\sqrt[9]{x} {\sqrt[9]{x}}^{8}}$

Step 3.4.4.4.2

Raise $\sqrt[9]{x}$ to the power of $1$ .

$y = \frac{{\sqrt[9]{x}}^{8}}{{\sqrt[9]{x}}^{1} {\sqrt[9]{x}}^{8}}$

Step 3.4.4.4.3

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

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

Step 3.4.4.4.4

Add $1$ and $8$ .

$y = \frac{{\sqrt[9]{x}}^{8}}{{\sqrt[9]{x}}^{9}}$

Step 3.4.4.4.5

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

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Step 3.4.4.4.5.1

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

$y = \frac{{\sqrt[9]{x}}^{8}}{{(x^{\frac{1}{9}})}^{9}}$

Step 3.4.4.4.5.2

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

$y = \frac{{\sqrt[9]{x}}^{8}}{x^{\frac{1}{9} \cdot 9}}$

Step 3.4.4.4.5.3

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

$y = \frac{{\sqrt[9]{x}}^{8}}{x^{\frac{9}{9}}}$

Step 3.4.4.4.5.4

Cancel the common factor of $9$ .

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Step 3.4.4.4.5.4.1

Cancel the common factor.

$y = \frac{{\sqrt[9]{x}}^{8}}{x^{\frac{9}{9}}}$

Step 3.4.4.4.5.4.2

Rewrite the expression.

$y = \frac{{\sqrt[9]{x}}^{8}}{x^{1}}$

Step 3.4.4.4.5.5

Simplify.

$y = \frac{{\sqrt[9]{x}}^{8}}{x}$

Step 3.4.4.5

Rewrite ${\sqrt[9]{x}}^{8}$ as $\sqrt[9]{x^{8}}$ .

$y = \frac{\sqrt[9]{x^{8}}}{x}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[9]{x^{8}}}{x}$

Step 5

Set up the composite result function.

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

Step 6

Evaluate $f^{- 1} (\frac{1}{x^{9}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{x^{9}}) = \frac{\sqrt[9]{{(\frac{1}{x^{9}})}^{8}}}{\frac{1}{x^{9}}}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{1}{x^{9}}) = \sqrt[9]{{(\frac{1}{x^{9}})}^{8}} x^{9}$

Step 8

Apply the product rule to $\frac{1}{x^{9}}$ .

$f^{- 1} (\frac{1}{x^{9}}) = \sqrt[9]{\frac{1^{8}}{{(x^{9})}^{8}}} x^{9}$

Step 9

One to any power is one.

$f^{- 1} (\frac{1}{x^{9}}) = \sqrt[9]{\frac{1}{{(x^{9})}^{8}}} x^{9}$

Step 10

Multiply the exponents in ${(x^{9})}^{8}$ .

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Step 10.1

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

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

Step 10.2

Multiply $9$ by $8$ .

$f^{- 1} (\frac{1}{x^{9}}) = \sqrt[9]{\frac{1}{x^{72}}} x^{9}$

Step 11

Rewrite $1$ as $1^{9}$ .

$f^{- 1} (\frac{1}{x^{9}}) = \sqrt[9]{\frac{1^{9}}{x^{72}}} x^{9}$

Step 12

Rewrite $x^{72}$ as ${(x^{8})}^{9}$ .

$f^{- 1} (\frac{1}{x^{9}}) = \sqrt[9]{\frac{1^{9}}{{(x^{8})}^{9}}} x^{9}$

Step 13

Rewrite $\frac{1^{9}}{{(x^{8})}^{9}}$ as ${(\frac{1}{x^{8}})}^{9}$ .

$f^{- 1} (\frac{1}{x^{9}}) = \sqrt[9]{{(\frac{1}{x^{8}})}^{9}} x^{9}$

Step 14

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

$f^{- 1} (\frac{1}{x^{9}}) = \frac{1}{x^{8}} \cdot x^{9}$

Step 15

Cancel the common factor of $x^{8}$ .

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Step 15.1

Factor $x^{8}$ out of $x^{9}$ .

$f^{- 1} (\frac{1}{x^{9}}) = \frac{1}{x^{8}} \cdot (x^{8} x)$

Step 15.2

Cancel the common factor.

$f^{- 1} (\frac{1}{x^{9}}) = \frac{1}{x^{8}} \cdot (x^{8} x)$

Step 15.3

Rewrite the expression.

$f^{- 1} (\frac{1}{x^{9}}) = x$