Basic Math Examples

$\frac{64 - y^{2}}{y^{2} + 6 y - 16} \cdot \frac{y^{2} + 17 y + 70}{y^{2} - y - 56}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $64$ as $8^{2}$ .

$\frac{8^{2} - y^{2}}{y^{2} + 6 y - 16} \cdot \frac{y^{2} + 17 y + 70}{y^{2} - y - 56}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = y$ .

$\frac{(8 + y) (8 - y)}{y^{2} + 6 y - 16} \cdot \frac{y^{2} + 17 y + 70}{y^{2} - y - 56}$

Step 2

Factor $y^{2} + 6 y - 16$ using the AC method.

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Step 2.1

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 $- 16$ and whose sum is $6$ .

$- 2, 8$

Step 2.2

Write the factored form using these integers.

$\frac{(8 + y) (8 - y)}{(y - 2) (y + 8)} \cdot \frac{y^{2} + 17 y + 70}{y^{2} - y - 56}$

Step 3

Factor $y^{2} + 17 y + 70$ using the AC method.

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Step 3.1

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 $70$ and whose sum is $17$ .

$7, 10$

Step 3.2

Write the factored form using these integers.

$\frac{(8 + y) (8 - y)}{(y - 2) (y + 8)} \cdot \frac{(y + 7) (y + 10)}{y^{2} - y - 56}$

Step 4

Factor $y^{2} - y - 56$ 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 $- 56$ and whose sum is $- 1$ .

$- 8, 7$

Step 4.2

Write the factored form using these integers.

$\frac{(8 + y) (8 - y)}{(y - 2) (y + 8)} \cdot \frac{(y + 7) (y + 10)}{(y - 8) (y + 7)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $8 + y$ and $y + 8$ .

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Step 5.1.1

Reorder terms.

$\frac{(y + 8) (8 - y)}{(y - 2) (y + 8)} \cdot \frac{(y + 7) (y + 10)}{(y - 8) (y + 7)}$

Step 5.1.2

Cancel the common factor.

$\frac{(y + 8) (8 - y)}{(y - 2) (y + 8)} \cdot \frac{(y + 7) (y + 10)}{(y - 8) (y + 7)}$

Step 5.1.3

Rewrite the expression.

$\frac{8 - y}{y - 2} \cdot \frac{(y + 7) (y + 10)}{(y - 8) (y + 7)}$

Step 5.2

Cancel the common factor of $y + 7$ .

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Step 5.2.1

Cancel the common factor.

$\frac{8 - y}{y - 2} \cdot \frac{(y + 7) (y + 10)}{(y - 8) (y + 7)}$

Step 5.2.2

Rewrite the expression.

$\frac{8 - y}{y - 2} \cdot \frac{y + 10}{y - 8}$

Step 5.3

Multiply $\frac{8 - y}{y - 2}$ by $\frac{y + 10}{y - 8}$ .

$\frac{(8 - y) (y + 10)}{(y - 2) (y - 8)}$

Step 5.4

Cancel the common factor of $8 - y$ and $y - 8$ .

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Step 5.4.1

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{(- 1 (- 8) - y) (y + 10)}{(y - 2) (y - 8)}$

Step 5.4.2

Factor $- 1$ out of $- y$ .

$\frac{(- 1 (- 8) - (y)) (y + 10)}{(y - 2) (y - 8)}$

Step 5.4.3

Factor $- 1$ out of $- 1 (- 8) - (y)$ .

$\frac{- 1 (- 8 + y) (y + 10)}{(y - 2) (y - 8)}$

Step 5.4.4

Reorder terms.

$\frac{- 1 (y - 8) (y + 10)}{(y - 2) (y - 8)}$

Step 5.4.5

Cancel the common factor.

$\frac{- 1 (y - 8) (y + 10)}{(y - 2) (y - 8)}$

Step 5.4.6

Rewrite the expression.

$\frac{- 1 (y + 10)}{y - 2}$

Step 5.5

Move the negative in front of the fraction.

$- \frac{y + 10}{y - 2}$