Basic Math Examples

$\frac{a^{2} - a - 6}{a^{2} - 81} \div \frac{a^{2} - 7 a - 18}{4 a + 36}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{a^{2} - a - 6}{a^{2} - 81} \cdot \frac{4 a + 36}{a^{2} - 7 a - 18}$

Step 2

Factor $a^{2} - a - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(a - 3) (a + 2)}{a^{2} - 81} \cdot \frac{4 a + 36}{a^{2} - 7 a - 18}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $81$ as $9^{2}$ .

$\frac{(a - 3) (a + 2)}{a^{2} - 9^{2}} \cdot \frac{4 a + 36}{a^{2} - 7 a - 18}$

Step 3.2

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 $b = 9$ .

$\frac{(a - 3) (a + 2)}{(a + 9) (a - 9)} \cdot \frac{4 a + 36}{a^{2} - 7 a - 18}$

Step 4

Factor $4$ out of $4 a + 36$ .

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Step 4.1

Factor $4$ out of $4 a$ .

$\frac{(a - 3) (a + 2)}{(a + 9) (a - 9)} \cdot \frac{4 (a) + 36}{a^{2} - 7 a - 18}$

Step 4.2

Factor $4$ out of $36$ .

$\frac{(a - 3) (a + 2)}{(a + 9) (a - 9)} \cdot \frac{4 a + 4 \cdot 9}{a^{2} - 7 a - 18}$

Step 4.3

Factor $4$ out of $4 a + 4 \cdot 9$ .

$\frac{(a - 3) (a + 2)}{(a + 9) (a - 9)} \cdot \frac{4 (a + 9)}{a^{2} - 7 a - 18}$

Step 5

Factor $a^{2} - 7 a - 18$ using the AC method.

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Step 5.1

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 $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Step 5.2

Write the factored form using these integers.

$\frac{(a - 3) (a + 2)}{(a + 9) (a - 9)} \cdot \frac{4 (a + 9)}{(a - 9) (a + 2)}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $a + 2$ .

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Step 6.1.1

Factor $a + 2$ out of $(a - 3) (a + 2)$ .

$\frac{(a + 2) (a - 3)}{(a + 9) (a - 9)} \cdot \frac{4 (a + 9)}{(a - 9) (a + 2)}$

Step 6.1.2

Factor $a + 2$ out of $(a - 9) (a + 2)$ .

$\frac{(a + 2) (a - 3)}{(a + 9) (a - 9)} \cdot \frac{4 (a + 9)}{(a + 2) (a - 9)}$

Step 6.1.3

Cancel the common factor.

$\frac{(a + 2) (a - 3)}{(a + 9) (a - 9)} \cdot \frac{4 (a + 9)}{(a + 2) (a - 9)}$

Step 6.1.4

Rewrite the expression.

$\frac{a - 3}{(a + 9) (a - 9)} \cdot \frac{4 (a + 9)}{a - 9}$

Step 6.2

Cancel the common factor of $a + 9$ .

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Step 6.2.1

Factor $a + 9$ out of $4 (a + 9)$ .

$\frac{a - 3}{(a + 9) (a - 9)} \cdot \frac{(a + 9) \cdot 4}{a - 9}$

Step 6.2.2

Cancel the common factor.

$\frac{a - 3}{(a + 9) (a - 9)} \cdot \frac{(a + 9) \cdot 4}{a - 9}$

Step 6.2.3

Rewrite the expression.

$\frac{a - 3}{a - 9} \cdot \frac{4}{a - 9}$

Step 6.3

Multiply $\frac{a - 3}{a - 9}$ by $\frac{4}{a - 9}$ .

$\frac{(a - 3) \cdot 4}{(a - 9) (a - 9)}$

Step 7

Raise $a - 9$ to the power of $1$ .

$\frac{(a - 3) \cdot 4}{{(a - 9)}^{1} (a - 9)}$

Step 8

Raise $a - 9$ to the power of $1$ .

$\frac{(a - 3) \cdot 4}{{(a - 9)}^{1} {(a - 9)}^{1}}$

Step 9

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

$\frac{(a - 3) \cdot 4}{{(a - 9)}^{1 + 1}}$

Step 10

Add $1$ and $1$ .

$\frac{(a - 3) \cdot 4}{{(a - 9)}^{2}}$

Step 11

Move $4$ to the left of $a - 3$ .

$\frac{4 (a - 3)}{{(a - 9)}^{2}}$