Evaluate ((y^4-16)/(y+2))÷((y^2+4)/5)

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

To divide by a fraction, multiply by its reciprocal.

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

Simplify the numerator.

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

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

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

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

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 $b = 4$ .

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

Simplify.

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

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

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 $b = 2$ .

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

Simplify terms.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Cancel the common factor of $y + 2$ .

Cancel the common factor.

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

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

$(y - 2) \cdot 5$

Apply the distributive property.

$y \cdot 5 - 2 \cdot 5$

Simplify the expression.

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

$5 \cdot y - 2 \cdot 5$

Multiply $- 2$ by $5$ .

$5 y - 10$

Evaluate ((y^4-16)/(y+2))÷((y^2+4)/5)

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