Basic Math Examples

$\frac{y^{2} - 16}{8} \div \frac{y^{2} - y - 12}{y^{2} - 8 y + 16}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} - 16}{8} \cdot \frac{y^{2} - 8 y + 16}{y^{2} - y - 12}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{y^{2} - 4^{2}}{8} \cdot \frac{y^{2} - 8 y + 16}{y^{2} - y - 12}$

Step 2.2

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 $b = 4$ .

$\frac{(y + 4) (y - 4)}{8} \cdot \frac{y^{2} - 8 y + 16}{y^{2} - y - 12}$

Step 3

Factor using the perfect square rule.

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Step 3.1

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

$\frac{(y + 4) (y - 4)}{8} \cdot \frac{y^{2} - 8 y + 4^{2}}{y^{2} - y - 12}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 y = 2 \cdot y \cdot 4$

Step 3.3

Rewrite the polynomial.

$\frac{(y + 4) (y - 4)}{8} \cdot \frac{y^{2} - 2 \cdot y \cdot 4 + 4^{2}}{y^{2} - y - 12}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 4$ .

$\frac{(y + 4) (y - 4)}{8} \cdot \frac{{(y - 4)}^{2}}{y^{2} - y - 12}$

Step 4

Factor $y^{2} - y - 12$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 4.2

Write the factored form using these integers.

$\frac{(y + 4) (y - 4)}{8} \cdot \frac{{(y - 4)}^{2}}{(y - 4) (y + 3)}$

Step 5

Combine.

$\frac{(y + 4) (y - 4) {(y - 4)}^{2}}{8 ((y - 4) (y + 3))}$

Step 6

Multiply $y - 4$ by ${(y - 4)}^{2}$ by adding the exponents.

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Step 6.1

Move ${(y - 4)}^{2}$ .

$\frac{(y + 4) ({(y - 4)}^{2} (y - 4))}{8 ((y - 4) (y + 3))}$

Step 6.2

Multiply ${(y - 4)}^{2}$ by $y - 4$ .

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Step 6.2.1

Raise $y - 4$ to the power of $1$ .

$\frac{(y + 4) ({(y - 4)}^{2} {(y - 4)}^{1})}{8 ((y - 4) (y + 3))}$

Step 6.2.2

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

$\frac{(y + 4) {(y - 4)}^{2 + 1}}{8 ((y - 4) (y + 3))}$

Step 6.3

Add $2$ and $1$ .

$\frac{(y + 4) {(y - 4)}^{3}}{8 ((y - 4) (y + 3))}$

Step 7

Cancel the common factor of ${(y - 4)}^{3}$ and $y - 4$ .

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Step 7.1

Factor $y - 4$ out of $(y + 4) {(y - 4)}^{3}$ .

$\frac{(y - 4) ((y + 4) {(y - 4)}^{2})}{8 ((y - 4) (y + 3))}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $y - 4$ out of $8 ((y - 4) (y + 3))$ .

$\frac{(y - 4) ((y + 4) {(y - 4)}^{2})}{(y - 4) (8 (y + 3))}$

Step 7.2.2

Cancel the common factor.

$\frac{(y - 4) ((y + 4) {(y - 4)}^{2})}{(y - 4) (8 (y + 3))}$

Step 7.2.3

Rewrite the expression.

$\frac{(y + 4) {(y - 4)}^{2}}{8 (y + 3)}$