Basic Math Examples

$\frac{3 a c + b c}{9 a^{2} + 6 a b + b^{2}} \cdot \frac{9 a^{2} - b^{2}}{3 a c - b c}$

Step 1

Factor $c$ out of $3 a c + b c$ .

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Step 1.1

Factor $c$ out of $3 a c$ .

$\frac{c (3 a) + b c}{9 a^{2} + 6 a b + b^{2}} \cdot \frac{9 a^{2} - b^{2}}{3 a c - b c}$

Step 1.2

Factor $c$ out of $b c$ .

$\frac{c (3 a) + c b}{9 a^{2} + 6 a b + b^{2}} \cdot \frac{9 a^{2} - b^{2}}{3 a c - b c}$

Step 1.3

Factor $c$ out of $c (3 a) + c b$ .

$\frac{c (3 a + b)}{9 a^{2} + 6 a b + b^{2}} \cdot \frac{9 a^{2} - b^{2}}{3 a c - b c}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $9 a^{2}$ as ${(3 a)}^{2}$ .

$\frac{c (3 a + b)}{{(3 a)}^{2} + 6 a b + b^{2}} \cdot \frac{9 a^{2} - b^{2}}{3 a c - b c}$

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.

$6 a b = 2 \cdot (3 a) \cdot b$

Step 2.3

Rewrite the polynomial.

$\frac{c (3 a + b)}{{(3 a)}^{2} + 2 \cdot (3 a) \cdot b + b^{2}} \cdot \frac{9 a^{2} - b^{2}}{3 a c - b c}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 3 a$ and $b = b$ .

$\frac{c (3 a + b)}{{(3 a + b)}^{2}} \cdot \frac{9 a^{2} - b^{2}}{3 a c - b c}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $9 a^{2}$ as ${(3 a)}^{2}$ .

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

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 $a = 3 a$ and $b = b$ .

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

Step 4

Simplify terms.

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Step 4.1

Factor $c$ out of $3 a c - b c$ .

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Step 4.1.1

Factor $c$ out of $3 a c$ .

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

Step 4.1.2

Factor $c$ out of $- b c$ .

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

Step 4.1.3

Factor $c$ out of $c (3 a) + c (- b)$ .

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

Step 4.2

Combine.

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

Step 4.3

Cancel the common factor of $c$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 5

Cancel the common factors.

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Step 5.1

Factor $3 a + b$ out of ${(3 a + b)}^{2} (3 a - b)$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $3 a + b$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{3 a - b}{3 a - b}$

Step 7

Cancel the common factor of $3 a - b$ .

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Step 7.1

Cancel the common factor.

$\frac{3 a - b}{3 a - b}$

Step 7.2

Rewrite the expression.

$1$