Basic Math Examples

$\frac{\frac{\frac{4 a^{2} - 1}{a^{2} - 4}}{2 a - 1}}{a + 2}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $4 a^{2}$ as ${(2 a)}^{2}$ .

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

Step 3.2

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

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

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a$ and $b = 1$ .

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

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 $b = 2$ .

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

Step 5

Cancel the common factor of $2 a - 1$ .

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

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

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Step 6.1

Multiply $\frac{2 a + 1}{(a + 2) (a - 2)}$ by $\frac{1}{a + 2}$ .

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

Step 6.2

Raise $a + 2$ to the power of $1$ .

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

Step 6.3

Raise $a + 2$ to the power of $1$ .

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

Step 6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.5

Add $1$ and $1$ .

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