Basic Math Examples

$\frac{2 a + 1}{10 a - 5} \div \frac{10 a}{4 a^{2} - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 a + 1}{10 a - 5} \cdot \frac{4 a^{2} - 1}{10 a}$

Step 2

Factor $5$ out of $10 a - 5$ .

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Step 2.1

Factor $5$ out of $10 a$ .

$\frac{2 a + 1}{5 (2 a) - 5} \cdot \frac{4 a^{2} - 1}{10 a}$

Step 2.2

Factor $5$ out of $- 5$ .

$\frac{2 a + 1}{5 (2 a) + 5 (- 1)} \cdot \frac{4 a^{2} - 1}{10 a}$

Step 2.3

Factor $5$ out of $5 (2 a) + 5 (- 1)$ .

$\frac{2 a + 1}{5 (2 a - 1)} \cdot \frac{4 a^{2} - 1}{10 a}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{2 a + 1}{5 (2 a - 1)} \cdot \frac{{(2 a)}^{2} - 1}{10 a}$

Step 3.2

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

$\frac{2 a + 1}{5 (2 a - 1)} \cdot \frac{{(2 a)}^{2} - 1^{2}}{10 a}$

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 $b = 1$ .

$\frac{2 a + 1}{5 (2 a - 1)} \cdot \frac{(2 a + 1) (2 a - 1)}{10 a}$

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

Factor $2 a - 1$ out of $5 (2 a - 1)$ .

$\frac{2 a + 1}{(2 a - 1) \cdot 5} \cdot \frac{(2 a + 1) (2 a - 1)}{10 a}$

Step 4.1.2

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

$\frac{2 a + 1}{(2 a - 1) \cdot 5} \cdot \frac{(2 a - 1) (2 a + 1)}{10 a}$

Step 4.1.3

Cancel the common factor.

$\frac{2 a + 1}{(2 a - 1) \cdot 5} \cdot \frac{(2 a - 1) (2 a + 1)}{10 a}$

Step 4.1.4

Rewrite the expression.

$\frac{2 a + 1}{5} \cdot \frac{2 a + 1}{10 a}$

Step 4.2

Multiply $\frac{2 a + 1}{5}$ by $\frac{2 a + 1}{10 a}$ .

$\frac{(2 a + 1) (2 a + 1)}{5 (10 a)}$

Step 5

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

$\frac{{(2 a + 1)}^{1} (2 a + 1)}{5 (10 a)}$

Step 6

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

$\frac{{(2 a + 1)}^{1} {(2 a + 1)}^{1}}{5 (10 a)}$

Step 7

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

$\frac{{(2 a + 1)}^{1 + 1}}{5 (10 a)}$

Step 8

Add $1$ and $1$ .

$\frac{{(2 a + 1)}^{2}}{5 (10 a)}$

Step 9

Multiply $10$ by $5$ .

$\frac{{(2 a + 1)}^{2}}{50 a}$