Pre-Algebra Examples

$\frac{s^{2} - 16}{9 s + 36} \cdot \frac{27 s + 108}{3 s^{2} - 24 s + 48}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{s^{2} - 4^{2}}{9 s + 36} \cdot \frac{27 s + 108}{3 s^{2} - 24 s + 48}$

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 $a = s$ and $b = 4$ .

$\frac{(s + 4) (s - 4)}{9 s + 36} \cdot \frac{27 s + 108}{3 s^{2} - 24 s + 48}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $9$ out of $9 s + 36$ .

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Step 2.1.1

Factor $9$ out of $9 s$ .

$\frac{(s + 4) (s - 4)}{9 (s) + 36} \cdot \frac{27 s + 108}{3 s^{2} - 24 s + 48}$

Step 2.1.2

Factor $9$ out of $36$ .

$\frac{(s + 4) (s - 4)}{9 s + 9 \cdot 4} \cdot \frac{27 s + 108}{3 s^{2} - 24 s + 48}$

Step 2.1.3

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

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 s + 108}{3 s^{2} - 24 s + 48}$

Step 2.2

Factor $27$ out of $27 s + 108$ .

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Step 2.2.1

Factor $27$ out of $27 s$ .

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s) + 108}{3 s^{2} - 24 s + 48}$

Step 2.2.2

Factor $27$ out of $108$ .

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 s + 27 \cdot 4}{3 s^{2} - 24 s + 48}$

Step 2.2.3

Factor $27$ out of $27 s + 27 \cdot 4$ .

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 s^{2} - 24 s + 48}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $3$ out of $3 s^{2} - 24 s + 48$ .

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Step 3.1.1

Factor $3$ out of $3 s^{2}$ .

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 (s^{2}) - 24 s + 48}$

Step 3.1.2

Factor $3$ out of $- 24 s$ .

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 (s^{2}) + 3 (- 8 s) + 48}$

Step 3.1.3

Factor $3$ out of $48$ .

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 s^{2} + 3 (- 8 s) + 3 \cdot 16}$

Step 3.1.4

Factor $3$ out of $3 s^{2} + 3 (- 8 s)$ .

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 (s^{2} - 8 s) + 3 \cdot 16}$

Step 3.1.5

Factor $3$ out of $3 (s^{2} - 8 s) + 3 \cdot 16$ .

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 (s^{2} - 8 s + 16)}$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

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

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 (s^{2} - 8 s + 4^{2})}$

Step 3.2.2

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

$8 s = 2 \cdot s \cdot 4$

Step 3.2.3

Rewrite the polynomial.

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 (s^{2} - 2 \cdot s \cdot 4 + 4^{2})}$

Step 3.2.4

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

$\frac{(s + 4) (s - 4)}{9 (s + 4)} \cdot \frac{27 (s + 4)}{3 {(s - 4)}^{2}}$

Step 4

Simplify terms.

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Step 4.1

Combine.

$\frac{(s + 4) (s - 4) (27 (s + 4))}{9 (s + 4) (3 {(s - 4)}^{2})}$

Step 4.2

Cancel the common factor of $s + 4$ .

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Step 4.2.1

Cancel the common factor.

$\frac{(s + 4) (s - 4) (27 (s + 4))}{9 (s + 4) (3 {(s - 4)}^{2})}$

Step 4.2.2

Rewrite the expression.

$\frac{(s - 4) (27 (s + 4))}{9 (3 {(s - 4)}^{2})}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $s - 4$ out of $9 (3 {(s - 4)}^{2})$ .

$\frac{(s - 4) (27 (s + 4))}{(s - 4) (9 (3 (s - 4)))}$

Step 5.2

Cancel the common factor.

$\frac{(s - 4) (27 (s + 4))}{(s - 4) (9 (3 (s - 4)))}$

Step 5.3

Rewrite the expression.

$\frac{27 (s + 4)}{9 (3 (s - 4))}$

Step 6

Cancel the common factor of $27$ and $9$ .

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Step 6.1

Factor $9$ out of $27 (s + 4)$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 7

Cancel the common factor of $3$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\frac{s + 4}{s - 4}$