Algebra Examples

$\frac{y^{4} - 16}{y + 2} \div \frac{y^{2} + 4}{5}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{4} - 16}{y + 2} \cdot \frac{5}{y^{2} + 4}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

$\frac{{(y^{2})}^{2} - 16}{y + 2} \cdot \frac{5}{y^{2} + 4}$

Step 2.2

Rewrite $16$ as $4^{2}$ .

$\frac{{(y^{2})}^{2} - 4^{2}}{y + 2} \cdot \frac{5}{y^{2} + 4}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y^{2}$ and $b = 4$ .

$\frac{(y^{2} + 4) (y^{2} - 4)}{y + 2} \cdot \frac{5}{y^{2} + 4}$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{(y^{2} + 4) (y^{2} - 2^{2})}{y + 2} \cdot \frac{5}{y^{2} + 4}$

Step 2.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 2$ .

$\frac{(y^{2} + 4) (y + 2) (y - 2)}{y + 2} \cdot \frac{5}{y^{2} + 4}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $y^{2} + 4$ .

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Step 3.1.1

Factor $y^{2} + 4$ out of $(y^{2} + 4) (y + 2) (y - 2)$ .

$\frac{(y^{2} + 4) ((y + 2) (y - 2))}{y + 2} \cdot \frac{5}{y^{2} + 4}$

Step 3.1.2

Cancel the common factor.

$\frac{(y^{2} + 4) ((y + 2) (y - 2))}{y + 2} \cdot \frac{5}{y^{2} + 4}$

Step 3.1.3

Rewrite the expression.

$\frac{(y + 2) (y - 2)}{y + 2} \cdot 5$

Step 3.2

Combine $\frac{(y + 2) (y - 2)}{y + 2}$ and $5$ .

$\frac{(y + 2) (y - 2) \cdot 5}{y + 2}$

Step 3.3

Cancel the common factor of $y + 2$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(y + 2) (y - 2) \cdot 5}{y + 2}$

Step 3.3.2

Divide $(y - 2) \cdot 5$ by $1$ .

$(y - 2) \cdot 5$

Step 3.4

Apply the distributive property.

$y \cdot 5 - 2 \cdot 5$

Step 3.5

Simplify the expression.

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Step 3.5.1

Move $5$ to the left of $y$ .

$5 \cdot y - 2 \cdot 5$

Step 3.5.2

Multiply $- 2$ by $5$ .

$5 y - 10$