Algebra Examples

$y = 3 x^{5} - 4$

Step 1

Interchange the variables.

$x = 3 y^{5} - 4$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $3 y^{5} - 4 = x$ .

$3 y^{5} - 4 = x$

Step 2.2

Add $4$ to both sides of the equation.

$3 y^{5} = x + 4$

Step 2.3

Divide each term in $3 y^{5} = x + 4$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 y^{5} = x + 4$ by $3$ .

$\frac{3 y^{5}}{3} = \frac{x}{3} + \frac{4}{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 y^{5}}{3} = \frac{x}{3} + \frac{4}{3}$

Step 2.3.2.1.2

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

$y^{5} = \frac{x}{3} + \frac{4}{3}$

Step 2.4

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

$y = \sqrt[5]{\frac{x}{3} + \frac{4}{3}}$

Step 2.5

Simplify $\sqrt[5]{\frac{x}{3} + \frac{4}{3}}$ .

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Step 2.5.1

Combine the numerators over the common denominator.

$y = \sqrt[5]{\frac{x + 4}{3}}$

Step 2.5.2

Rewrite $\sqrt[5]{\frac{x + 4}{3}}$ as $\frac{\sqrt[5]{x + 4}}{\sqrt[5]{3}}$ .

$y = \frac{\sqrt[5]{x + 4}}{\sqrt[5]{3}}$

Step 2.5.3

Multiply $\frac{\sqrt[5]{x + 4}}{\sqrt[5]{3}}$ by $\frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$ .

$y = \frac{\sqrt[5]{x + 4}}{\sqrt[5]{3}} \cdot \frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$

Step 2.5.4

Combine and simplify the denominator.

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Step 2.5.4.1

Multiply $\frac{\sqrt[5]{x + 4}}{\sqrt[5]{3}}$ by $\frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$ .

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{\sqrt[5]{3} {\sqrt[5]{3}}^{4}}$

Step 2.5.4.2

Raise $\sqrt[5]{3}$ to the power of $1$ .

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{1} {\sqrt[5]{3}}^{4}}$

Step 2.5.4.3

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

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{1 + 4}}$

Step 2.5.4.4

Add $1$ and $4$ .

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{5}}$

Step 2.5.4.5

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

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Step 2.5.4.5.1

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

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{{(3^{\frac{1}{5}})}^{5}}$

Step 2.5.4.5.2

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

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{3^{\frac{1}{5} \cdot 5}}$

Step 2.5.4.5.3

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

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{3^{\frac{5}{5}}}$

Step 2.5.4.5.4

Cancel the common factor of $5$ .

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Step 2.5.4.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{3^{\frac{5}{5}}}$

Step 2.5.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{3^{1}}$

Step 2.5.4.5.5

Evaluate the exponent.

$y = \frac{\sqrt[5]{x + 4} {\sqrt[5]{3}}^{4}}{3}$

Step 2.5.5

Simplify the numerator.

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Step 2.5.5.1

Rewrite ${\sqrt[5]{3}}^{4}$ as $\sqrt[5]{3^{4}}$ .

$y = \frac{\sqrt[5]{x + 4} \sqrt[5]{3^{4}}}{3}$

Step 2.5.5.2

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

$y = \frac{\sqrt[5]{x + 4} \sqrt[5]{81}}{3}$

Step 2.5.6

Simplify with factoring out.

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Step 2.5.6.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[5]{(x + 4) \cdot 81}}{3}$

Step 2.5.6.2

Reorder factors in $\frac{\sqrt[5]{(x + 4) \cdot 81}}{3}$ .

$y = \frac{\sqrt[5]{81 (x + 4)}}{3}$

Step 3

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

$f^{- 1} (x) = \frac{\sqrt[5]{81 (x + 4)}}{3}$

Step 4

Verify if $f^{- 1} (x) = \frac{\sqrt[5]{81 (x + 4)}}{3}$ is the inverse of $f (x) = 3 x^{5} - 4$ .

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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 $f^{- 1} (3 x^{5} - 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 x^{5} - 4) = \frac{\sqrt[5]{81 ((3 x^{5} - 4) + 4)}}{3}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Add $- 4$ and $4$ .

$f^{- 1} (3 x^{5} - 4) = \frac{\sqrt[5]{81 (3 x^{5} + 0)}}{3}$

Step 4.2.3.2

Add $3 x^{5}$ and $0$ .

$f^{- 1} (3 x^{5} - 4) = \frac{\sqrt[5]{81 \cdot (3 x^{5})}}{3}$

Step 4.2.3.3

Multiply $81$ by $3$ .

$f^{- 1} (3 x^{5} - 4) = \frac{\sqrt[5]{243 x^{5}}}{3}$

Step 4.2.3.4

Rewrite $243 x^{5}$ as ${(3 x)}^{5}$ .

$f^{- 1} (3 x^{5} - 4) = \frac{\sqrt[5]{{(3 x)}^{5}}}{3}$

Step 4.2.3.5

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

$f^{- 1} (3 x^{5} - 4) = \frac{3 x}{3}$

Step 4.2.4

Cancel the common factor of $3$ .

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Step 4.2.4.1

Cancel the common factor.

$f^{- 1} (3 x^{5} - 4) = \frac{3 x}{3}$

Step 4.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (3 x^{5} - 4) = x$

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 $f (\frac{\sqrt[5]{81 (x + 4)}}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 {(\frac{\sqrt[5]{81 (x + 4)}}{3})}^{5} - 4$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the product rule to $\frac{\sqrt[5]{81 (x + 4)}}{3}$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{{\sqrt[5]{81 (x + 4)}}^{5}}{3^{5}}) - 4$

Step 4.3.3.2

Simplify the numerator.

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Step 4.3.3.2.1

Rewrite ${\sqrt[5]{81 (x + 4)}}^{5}$ as $81 (x + 4)$ .

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Step 4.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{81 (x + 4)}$ as ${(81 (x + 4))}^{\frac{1}{5}}$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{{({(81 (x + 4))}^{\frac{1}{5}})}^{5}}{3^{5}}) - 4$

Step 4.3.3.2.1.2

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

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{{(81 (x + 4))}^{\frac{1}{5} \cdot 5}}{3^{5}}) - 4$

Step 4.3.3.2.1.3

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

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{{(81 (x + 4))}^{\frac{5}{5}}}{3^{5}}) - 4$

Step 4.3.3.2.1.4

Cancel the common factor of $5$ .

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Step 4.3.3.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{{(81 (x + 4))}^{\frac{5}{5}}}{3^{5}}) - 4$

Step 4.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 (x + 4)}{3^{5}}) - 4$

Step 4.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 (x + 4)}{3^{5}}) - 4$

Step 4.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 x + 81 \cdot 4}{3^{5}}) - 4$

Step 4.3.3.2.3

Multiply $81$ by $4$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 x + 324}{3^{5}}) - 4$

Step 4.3.3.2.4

Factor $81$ out of $81 x + 324$ .

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Step 4.3.3.2.4.1

Factor $81$ out of $81 x$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 (x) + 324}{3^{5}}) - 4$

Step 4.3.3.2.4.2

Factor $81$ out of $324$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 x + 81 \cdot 4}{3^{5}}) - 4$

Step 4.3.3.2.4.3

Factor $81$ out of $81 x + 81 \cdot 4$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 (x + 4)}{3^{5}}) - 4$

Step 4.3.3.3

Raise $3$ to the power of $5$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 (x + 4)}{243}) - 4$

Step 4.3.3.4

Cancel the common factor of $3$ .

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Step 4.3.3.4.1

Factor $3$ out of $243$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 (x + 4)}{3 (81)}) - 4$

Step 4.3.3.4.2

Cancel the common factor.

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = 3 (\frac{81 (x + 4)}{3 \cdot 81}) - 4$

Step 4.3.3.4.3

Rewrite the expression.

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = \frac{81 (x + 4)}{81} - 4$

Step 4.3.3.5

Cancel the common factor of $81$ .

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Step 4.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = \frac{81 (x + 4)}{81} - 4$

Step 4.3.3.5.2

Divide $x + 4$ by $1$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = x + 4 - 4$

Step 4.3.4

Combine the opposite terms in $x + 4 - 4$ .

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Step 4.3.4.1

Subtract $4$ from $4$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[5]{81 (x + 4)}}{3}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[5]{81 (x + 4)}}{3}$ is the inverse of $f (x) = 3 x^{5} - 4$ .

$f^{- 1} (x) = \frac{\sqrt[5]{81 (x + 4)}}{3}$