Basic Math Examples

$\frac{a^{3} + a^{2} + a + b}{3 a^{3} + 3 a} \cdot \frac{18 a^{2}}{9 a^{2} - 9 b^{2}}$

Step 1

Factor $3 a$ out of $3 a^{3} + 3 a$ .

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Step 1.1

Factor $3 a$ out of $3 a^{3}$ .

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2}) + 3 a} \cdot \frac{18 a^{2}}{9 a^{2} - 9 b^{2}}$

Step 1.2

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

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2}) + 3 a (1)} \cdot \frac{18 a^{2}}{9 a^{2} - 9 b^{2}}$

Step 1.3

Factor $3 a$ out of $3 a (a^{2}) + 3 a (1)$ .

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2} + 1)} \cdot \frac{18 a^{2}}{9 a^{2} - 9 b^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $9$ out of $9 a^{2} - 9 b^{2}$ .

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Step 2.1.1

Factor $9$ out of $9 a^{2}$ .

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2} + 1)} \cdot \frac{18 a^{2}}{9 (a^{2}) - 9 b^{2}}$

Step 2.1.2

Factor $9$ out of $- 9 b^{2}$ .

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2} + 1)} \cdot \frac{18 a^{2}}{9 (a^{2}) + 9 (- b^{2})}$

Step 2.1.3

Factor $9$ out of $9 (a^{2}) + 9 (- b^{2})$ .

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2} + 1)} \cdot \frac{18 a^{2}}{9 (a^{2} - b^{2})}$

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 $b = b$ .

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2} + 1)} \cdot \frac{18 a^{2}}{9 (a + b) (a - b)}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $3 a$ .

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Step 3.1.1

Factor $3 a$ out of $18 a^{2}$ .

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2} + 1)} \cdot \frac{3 a (6 a)}{9 (a + b) (a - b)}$

Step 3.1.2

Cancel the common factor.

$\frac{a^{3} + a^{2} + a + b}{3 a (a^{2} + 1)} \cdot \frac{3 a (6 a)}{9 (a + b) (a - b)}$

Step 3.1.3

Rewrite the expression.

$\frac{a^{3} + a^{2} + a + b}{a^{2} + 1} \cdot \frac{6 a}{9 (a + b) (a - b)}$

Step 3.2

Multiply $\frac{a^{3} + a^{2} + a + b}{a^{2} + 1}$ by $\frac{6 a}{9 (a + b) (a - b)}$ .

$\frac{(a^{3} + a^{2} + a + b) (6 a)}{(a^{2} + 1) (9 (a + b) (a - b))}$

Step 3.3

Cancel the common factor of $6$ and $9$ .

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Step 3.3.1

Factor $3$ out of $(a^{3} + a^{2} + a + b) (6 a)$ .

$\frac{3 ((a^{3} + a^{2} + a + b) (2 a))}{(a^{2} + 1) (9 (a + b) (a - b))}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $3$ out of $(a^{2} + 1) (9 (a + b) (a - b))$ .

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Move $2$ to the left of $a^{3} + a^{2} + a + b$ .

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

Step 3.4.2

Move $3$ to the left of $a^{2} + 1$ .

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

Step 3.4.3

Reorder factors in $\frac{2 (a^{3} + a^{2} + a + b) a}{3 (a^{2} + 1) (a + b) (a - b)}$ .

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