Basic Math Examples

$\frac{a^{2} + 7 a + 12}{a^{2} - 9} \div \frac{a^{2} - 7 a + 12}{a^{2} - 16}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{a^{2} + 7 a + 12}{a^{2} - 9} \cdot \frac{a^{2} - 16}{a^{2} - 7 a + 12}$

Step 2

Factor $a^{2} + 7 a + 12$ 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 $12$ and whose sum is $7$ .

$3, 4$

Step 2.2

Write the factored form using these integers.

$\frac{(a + 3) (a + 4)}{a^{2} - 9} \cdot \frac{a^{2} - 16}{a^{2} - 7 a + 12}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\frac{(a + 3) (a + 4)}{a^{2} - 3^{2}} \cdot \frac{a^{2} - 16}{a^{2} - 7 a + 12}$

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 $b = 3$ .

$\frac{(a + 3) (a + 4)}{(a + 3) (a - 3)} \cdot \frac{a^{2} - 16}{a^{2} - 7 a + 12}$

Step 4

Cancel the common factor of $a + 3$ .

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Step 4.1

Cancel the common factor.

$\frac{(a + 3) (a + 4)}{(a + 3) (a - 3)} \cdot \frac{a^{2} - 16}{a^{2} - 7 a + 12}$

Step 4.2

Rewrite the expression.

$\frac{a + 4}{a - 3} \cdot \frac{a^{2} - 16}{a^{2} - 7 a + 12}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{a + 4}{a - 3} \cdot \frac{a^{2} - 4^{2}}{a^{2} - 7 a + 12}$

Step 5.2

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 $b = 4$ .

$\frac{a + 4}{a - 3} \cdot \frac{(a + 4) (a - 4)}{a^{2} - 7 a + 12}$

Step 6

Factor $a^{2} - 7 a + 12$ using the AC method.

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Step 6.1

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 $- 7$ .

$- 4, - 3$

Step 6.2

Write the factored form using these integers.

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

Step 7

Cancel the common factor of $a - 4$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Multiply $\frac{a + 4}{a - 3} \cdot \frac{a + 4}{a - 3}$ .

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Step 8.1

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

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

Step 8.2

Raise $a + 4$ to the power of $1$ .

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

Step 8.3

Raise $a + 4$ to the power of $1$ .

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

Raise $a - 3$ to the power of $1$ .

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

Step 8.7

Raise $a - 3$ to the power of $1$ .

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

Step 8.8

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

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

Step 8.9

Add $1$ and $1$ .

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