Precalculus Examples

$y^{- 2} + 5 y = 0$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{y^{2}} + 5 y = 0$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$y^{2}, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$y^{2}$

Step 3

Multiply each term in $\frac{1}{y^{2}} + 5 y = 0$ by $y^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{y^{2}} + 5 y = 0$ by $y^{2}$ .

$\frac{1}{y^{2}} y^{2} + 5 y \cdot y^{2} = 0 y^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $y^{2}$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{1}{y^{2}} y^{2} + 5 y \cdot y^{2} = 0 y^{2}$

Step 3.2.1.1.2

Rewrite the expression.

$1 + 5 y \cdot y^{2} = 0 y^{2}$

Step 3.2.1.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 3.2.1.2.1

Move $y^{2}$ .

$1 + 5 (y^{2} y) = 0 y^{2}$

Step 3.2.1.2.2

Multiply $y^{2}$ by $y$ .

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Step 3.2.1.2.2.1

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

$1 + 5 (y^{2} y^{1}) = 0 y^{2}$

Step 3.2.1.2.2.2

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

$1 + 5 y^{2 + 1} = 0 y^{2}$

Step 3.2.1.2.3

Add $2$ and $1$ .

$1 + 5 y^{3} = 0 y^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $0$ by $y^{2}$ .

$1 + 5 y^{3} = 0$

Step 4

Solve the equation.

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Step 4.1

Subtract $1$ from both sides of the equation.

$5 y^{3} = - 1$

Step 4.2

Divide each term in $5 y^{3} = - 1$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 y^{3} = - 1$ by $5$ .

$\frac{5 y^{3}}{5} = \frac{- 1}{5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{5 y^{3}}{5} = \frac{- 1}{5}$

Step 4.2.2.1.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{- 1}{5}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$y^{3} = - \frac{1}{5}$

Step 4.3

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

$y = \sqrt[3]{- \frac{1}{5}}$

Step 4.4

Simplify $\sqrt[3]{- \frac{1}{5}}$ .

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Step 4.4.1

Rewrite $- \frac{1}{5}$ as ${({(- 1)}^{3})}^{3} \frac{1}{5}$ .

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Step 4.4.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 4.4.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 4.4.2

Pull terms out from under the radical.

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

Step 4.4.3

Raise $- 1$ to the power of $3$ .

$y = - \sqrt[3]{\frac{1}{5}}$

Step 4.4.4

Rewrite $\sqrt[3]{\frac{1}{5}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{5}}$ .

$y = - \frac{\sqrt[3]{1}}{\sqrt[3]{5}}$

Step 4.4.5

Any root of $1$ is $1$ .

$y = - \frac{1}{\sqrt[3]{5}}$

Step 4.4.6

Multiply $\frac{1}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$y = - (\frac{1}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}})$

Step 4.4.7

Combine and simplify the denominator.

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Step 4.4.7.1

Multiply $\frac{1}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$y = - \frac{{\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 4.4.7.2

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

$y = - \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

Step 4.4.7.3

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

$y = - \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Step 4.4.7.4

Add $1$ and $2$ .

$y = - \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Step 4.4.7.5

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

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Step 4.4.7.5.1

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

$y = - \frac{{\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

Step 4.4.7.5.2

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

$y = - \frac{{\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

Step 4.4.7.5.3

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

$y = - \frac{{\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 4.4.7.5.4

Cancel the common factor of $3$ .

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Step 4.4.7.5.4.1

Cancel the common factor.

$y = - \frac{{\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 4.4.7.5.4.2

Rewrite the expression.

$y = - \frac{{\sqrt[3]{5}}^{2}}{5^{1}}$

Step 4.4.7.5.5

Evaluate the exponent.

$y = - \frac{{\sqrt[3]{5}}^{2}}{5}$

Step 4.4.8

Simplify the numerator.

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Step 4.4.8.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$y = - \frac{\sqrt[3]{5^{2}}}{5}$

Step 4.4.8.2

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

$y = - \frac{\sqrt[3]{25}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{\sqrt[3]{25}}{5}$

Decimal Form:

$y = - 0.58480354 \dots$