Pre-Algebra Examples

$\frac{a^{2} - 36}{4 a^{2} + 4 a b} \cdot \frac{a^{2} + 2 a + a b + 2 b}{a^{2} + 8 a + 12}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $36$ as $6^{2}$ .

$\frac{a^{2} - 6^{2}}{4 a^{2} + 4 a b} \cdot \frac{a^{2} + 2 a + a b + 2 b}{a^{2} + 8 a + 12}$

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 $a = a$ and $b = 6$ .

$\frac{(a + 6) (a - 6)}{4 a^{2} + 4 a b} \cdot \frac{a^{2} + 2 a + a b + 2 b}{a^{2} + 8 a + 12}$

Step 2

Factor $4 a$ out of $4 a^{2} + 4 a b$ .

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Step 2.1

Factor $4 a$ out of $4 a^{2}$ .

$\frac{(a + 6) (a - 6)}{4 a (a) + 4 a b} \cdot \frac{a^{2} + 2 a + a b + 2 b}{a^{2} + 8 a + 12}$

Step 2.2

Factor $4 a$ out of $4 a b$ .

$\frac{(a + 6) (a - 6)}{4 a (a) + 4 a (b)} \cdot \frac{a^{2} + 2 a + a b + 2 b}{a^{2} + 8 a + 12}$

Step 2.3

Factor $4 a$ out of $4 a (a) + 4 a (b)$ .

$\frac{(a + 6) (a - 6)}{4 a (a + b)} \cdot \frac{a^{2} + 2 a + a b + 2 b}{a^{2} + 8 a + 12}$

Step 3

Simplify the numerator.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

$\frac{(a + 6) (a - 6)}{4 a (a + b)} \cdot \frac{(a^{2} + 2 a) + a b + 2 b}{a^{2} + 8 a + 12}$

Step 3.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(a + 6) (a - 6)}{4 a (a + b)} \cdot \frac{a (a + 2) + b (a + 2)}{a^{2} + 8 a + 12}$

Step 3.2

Factor the polynomial by factoring out the greatest common factor, $a + 2$ .

$\frac{(a + 6) (a - 6)}{4 a (a + b)} \cdot \frac{(a + 2) (a + b)}{a^{2} + 8 a + 12}$

Step 4

Factor $a^{2} + 8 a + 12$ 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 $8$ .

$2, 6$

Step 4.2

Write the factored form using these integers.

$\frac{(a + 6) (a - 6)}{4 a (a + b)} \cdot \frac{(a + 2) (a + b)}{(a + 2) (a + 6)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $a + 6$ .

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Step 5.1.1

Factor $a + 6$ out of $(a + 2) (a + 6)$ .

$\frac{(a + 6) (a - 6)}{4 a (a + b)} \cdot \frac{(a + 2) (a + b)}{(a + 6) (a + 2)}$

Step 5.1.2

Cancel the common factor.

$\frac{(a + 6) (a - 6)}{4 a (a + b)} \cdot \frac{(a + 2) (a + b)}{(a + 6) (a + 2)}$

Step 5.1.3

Rewrite the expression.

$\frac{a - 6}{4 a (a + b)} \cdot \frac{(a + 2) (a + b)}{a + 2}$

Step 5.2

Cancel the common factor of $a + b$ .

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Step 5.2.1

Factor $a + b$ out of $4 a (a + b)$ .

$\frac{a - 6}{(a + b) (4 a)} \cdot \frac{(a + 2) (a + b)}{a + 2}$

Step 5.2.2

Factor $a + b$ out of $(a + 2) (a + b)$ .

$\frac{a - 6}{(a + b) (4 a)} \cdot \frac{(a + b) (a + 2)}{a + 2}$

Step 5.2.3

Cancel the common factor.

$\frac{a - 6}{(a + b) (4 a)} \cdot \frac{(a + b) (a + 2)}{a + 2}$

Step 5.2.4

Rewrite the expression.

$\frac{a - 6}{4 a} \cdot \frac{a + 2}{a + 2}$

Step 5.3

Multiply $\frac{a - 6}{4 a}$ by $\frac{a + 2}{a + 2}$ .

$\frac{(a - 6) (a + 2)}{4 a (a + 2)}$

Step 5.4

Cancel the common factor of $a + 2$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(a - 6) (a + 2)}{4 a (a + 2)}$

Step 5.4.2

Rewrite the expression.

$\frac{a - 6}{4 a}$