Basic Math Examples

$\frac{3 (a + 2)}{a^{2} - 1} \div \frac{a^{2} - 4}{2 a^{2} + 2 a}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 (a + 2)}{a^{2} - 1} \cdot \frac{2 a^{2} + 2 a}{a^{2} - 4}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$\frac{3 (a + 2)}{a^{2} - 1^{2}} \cdot \frac{2 a^{2} + 2 a}{a^{2} - 4}$

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 $b = 1$ .

$\frac{3 (a + 2)}{(a + 1) (a - 1)} \cdot \frac{2 a^{2} + 2 a}{a^{2} - 4}$

Step 3

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

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Step 3.1

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

$\frac{3 (a + 2)}{(a + 1) (a - 1)} \cdot \frac{2 a (a) + 2 a}{a^{2} - 4}$

Step 3.2

Factor $2 a$ out of $2 a$ .

$\frac{3 (a + 2)}{(a + 1) (a - 1)} \cdot \frac{2 a (a) + 2 a (1)}{a^{2} - 4}$

Step 3.3

Factor $2 a$ out of $2 a (a) + 2 a (1)$ .

$\frac{3 (a + 2)}{(a + 1) (a - 1)} \cdot \frac{2 a (a + 1)}{a^{2} - 4}$

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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 $b = 2$ .

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $a + 2$ .

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Step 5.1.1

Factor $a + 2$ out of $3 (a + 2)$ .

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $a + 1$ .

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Step 5.2.1

Factor $a + 1$ out of $2 a (a + 1)$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{3}{a - 1} \cdot \frac{2 a}{a - 2}$

Step 5.3

Multiply $\frac{3}{a - 1}$ by $\frac{2 a}{a - 2}$ .

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

Step 5.4

Multiply $2$ by $3$ .

$\frac{6 a}{(a - 1) (a - 2)}$