Basic Math Examples

$\frac{3 a^{2}}{a^{2} + 8 a + 16} \cdot \frac{a^{2} - 16}{3 a}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$\frac{3 a^{2}}{a^{2} + 8 a + 4^{2}} \cdot \frac{a^{2} - 16}{3 a}$

Step 1.2

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 1.3

Rewrite the polynomial.

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

Step 1.4

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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 $b = 4$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine.

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

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $a^{2}$ and $a$ .

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Step 3.3.1

Factor $a$ out of $a^{2} ((a + 4) (a - 4))$ .

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

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

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

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