Basic Math Examples

$\frac{\frac{4}{a^{2} + 8 a + 16}}{\frac{7}{a^{2} - 16}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{4}{a^{2} + 8 a + 16} \cdot \frac{a^{2} - 16}{7}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{4}{a^{2} + 8 a + 4^{2}} \cdot \frac{a^{2} - 16}{7}$

Step 2.2

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

$8 a = 2 \cdot a \cdot 4$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

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 $b = 4$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

Cancel the common factor of $a + 4$ and ${(a + 4)}^{2}$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $a + 4$ out of ${(a + 4)}^{2} \cdot 7$ .

$\frac{(a + 4) (4 (a - 4))}{(a + 4) ((a + 4) \cdot 7)}$

Step 4.2.2.2

Cancel the common factor.

$\frac{(a + 4) (4 (a - 4))}{(a + 4) ((a + 4) \cdot 7)}$

Step 4.2.2.3

Rewrite the expression.

$\frac{4 (a - 4)}{(a + 4) \cdot 7}$

Step 4.3

Move $7$ to the left of $a + 4$ .

$\frac{4 (a - 4)}{7 (a + 4)}$