Finite Math Examples

$f (x) = \sqrt[3]{x^{7}}$

Step 1

Write $f (x) = \sqrt[3]{x^{7}}$ as an equation.

$y = \sqrt[3]{x^{7}}$

Step 2

Interchange the variables.

$x = \sqrt[3]{y^{7}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[3]{y^{7}} = x$ .

$\sqrt[3]{y^{7}} = x$

Step 3.2

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y^{7}}}^{3} = x^{3}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{7}}$ as $y^{\frac{7}{3}}$ .

${(y^{\frac{7}{3}})}^{3} = x^{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Multiply the exponents in ${(y^{\frac{7}{3}})}^{3}$ .

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Step 3.3.2.1.1

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

$y^{\frac{7}{3} \cdot 3} = x^{3}$

Step 3.3.2.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.1

Cancel the common factor.

$y^{\frac{7}{3} \cdot 3} = x^{3}$

Step 3.3.2.1.2.2

Rewrite the expression.

$y^{7} = x^{3}$

Step 3.4

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

$y = \sqrt[7]{x^{3}}$

Step 4

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

$f^{- 1} (x) = \sqrt[7]{x^{3}}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[7]{x^{3}}$ is the inverse of $f (x) = \sqrt[3]{x^{7}}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\sqrt[3]{x^{7}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{{(\sqrt[3]{x^{7}})}^{3}}$

Step 5.2.3

Rewrite $x^{7}$ as ${(x^{2})}^{3} x$ .

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Step 5.2.3.1

Factor out $x^{6}$ .

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{{\sqrt[3]{x^{6} x}}^{3}}$

Step 5.2.3.2

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{{\sqrt[3]{{(x^{2})}^{3} x}}^{3}}$

Step 5.2.4

Pull terms out from under the radical.

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{{(x^{2} \sqrt[3]{x})}^{3}}$

Step 5.2.5

Apply basic rules of exponents.

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Step 5.2.5.1

Apply the product rule to $x^{2} \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{{(x^{2})}^{3} {\sqrt[3]{x}}^{3}}$

Step 5.2.5.2

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 5.2.5.2.1

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

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{2 \cdot 3} {\sqrt[3]{x}}^{3}}$

Step 5.2.5.2.2

Multiply $2$ by $3$ .

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6} {\sqrt[3]{x}}^{3}}$

Step 5.2.6

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

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Step 5.2.6.1

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

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6} {(x^{\frac{1}{3}})}^{3}}$

Step 5.2.6.2

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

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6} x^{\frac{1}{3} \cdot 3}}$

Step 5.2.6.3

Combine $\frac{1}{3}$ and $3$ .

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6} x^{\frac{3}{3}}}$

Step 5.2.6.4

Cancel the common factor of $3$ .

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Step 5.2.6.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6} x^{\frac{3}{3}}}$

Step 5.2.6.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6} x}$

Step 5.2.6.5

Simplify.

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6} x}$

Step 5.2.7

Multiply $x^{6}$ by $x$ by adding the exponents.

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Step 5.2.7.1

Multiply $x^{6}$ by $x$ .

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Step 5.2.7.1.1

Raise $x$ to the power of $1$ .

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6} x}$

Step 5.2.7.1.2

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

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{6 + 1}}$

Step 5.2.7.2

Add $6$ and $1$ .

$f^{- 1} (\sqrt[3]{x^{7}}) = \sqrt[7]{x^{7}}$

Step 5.2.8

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

$f^{- 1} (\sqrt[3]{x^{7}}) = x$

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 $f (\sqrt[7]{x^{3}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{{(\sqrt[7]{x^{3}})}^{7}}$

Step 5.3.3

Rewrite ${\sqrt[7]{x^{3}}}^{7}$ as $x^{3}$ .

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Step 5.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{3}}$ as $x^{\frac{3}{7}}$ .

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{{(x^{\frac{3}{7}})}^{7}}$

Step 5.3.3.2

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

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{x^{\frac{3}{7} \cdot 7}}$

Step 5.3.3.3

Combine $\frac{3}{7}$ and $7$ .

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{x^{\frac{3 \cdot 7}{7}}}$

Step 5.3.3.4

Multiply $3$ by $7$ .

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{x^{\frac{21}{7}}}$

Step 5.3.3.5

Cancel the common factor of $21$ and $7$ .

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Step 5.3.3.5.1

Factor $7$ out of $21$ .

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{x^{\frac{7 \cdot 3}{7}}}$

Step 5.3.3.5.2

Cancel the common factors.

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Step 5.3.3.5.2.1

Factor $7$ out of $7$ .

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{x^{\frac{7 \cdot 3}{7 (1)}}}$

Step 5.3.3.5.2.2

Cancel the common factor.

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{x^{\frac{7 \cdot 3}{7 \cdot 1}}}$

Step 5.3.3.5.2.3

Rewrite the expression.

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{x^{\frac{3}{1}}}$

Step 5.3.3.5.2.4

Divide $3$ by $1$ .

$f (\sqrt[7]{x^{3}}) = \sqrt[3]{x^{3}}$

Step 5.3.4

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

$f (\sqrt[7]{x^{3}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[7]{x^{3}}$ is the inverse of $f (x) = \sqrt[3]{x^{7}}$ .

$f^{- 1} (x) = \sqrt[7]{x^{3}}$