Basic Math Examples

$\frac{5}{m^{2} - 4 n^{2}} + \frac{4}{m - 2 n}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{5}{m^{2} - {(2 n)}^{2}} + \frac{4}{m - 2 n}$

Step 1.2

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

$\frac{5}{(m + 2 n) (m - (2 n))} + \frac{4}{m - 2 n}$

Step 1.3

Multiply $2$ by $- 1$ .

$\frac{5}{(m + 2 n) (m - 2 n)} + \frac{4}{m - 2 n}$

Step 2

To write $\frac{4}{m - 2 n}$ as a fraction with a common denominator, multiply by $\frac{m + 2 n}{m + 2 n}$ .

$\frac{5}{(m + 2 n) (m - 2 n)} + \frac{4}{m - 2 n} \cdot \frac{m + 2 n}{m + 2 n}$

Step 3

Write each expression with a common denominator of $(m + 2 n) (m - 2 n)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{4}{m - 2 n}$ by $\frac{m + 2 n}{m + 2 n}$ .

$\frac{5}{(m + 2 n) (m - 2 n)} + \frac{4 (m + 2 n)}{(m - 2 n) (m + 2 n)}$

Step 3.2

Reorder the factors of $(m - 2 n) (m + 2 n)$ .

$\frac{5}{(m + 2 n) (m - 2 n)} + \frac{4 (m + 2 n)}{(m + 2 n) (m - 2 n)}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 + 4 (m + 2 n)}{(m + 2 n) (m - 2 n)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{5 + 4 m + 4 (2 n)}{(m + 2 n) (m - 2 n)}$

Step 5.2

Multiply $2$ by $4$ .

$\frac{5 + 4 m + 8 n}{(m + 2 n) (m - 2 n)}$