Simplify ((3m-6n)/(5n))/((m^2-4n^2)/(10mn))

$\frac{\frac{3 m - 6 n}{5 n}}{\frac{m^{2} - 4 n^{2}}{10 m n}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 m - 6 n}{5 n} \cdot \frac{10 m n}{m^{2} - 4 n^{2}}$

Step 2

Cancel the common factor of $5 n$ .

Factor $5 n$ out of $10 m n$ .

$\frac{3 m - 6 n}{5 n} \cdot \frac{5 n (2 m)}{m^{2} - 4 n^{2}}$

Cancel the common factor.

$\frac{3 m - 6 n}{5 n} \cdot \frac{5 n (2 m)}{m^{2} - 4 n^{2}}$

Rewrite the expression.

$(3 m - 6 n) \frac{2 m}{m^{2} - 4 n^{2}}$

Step 3

Simplify the denominator.

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

$(3 m - 6 n) \frac{2 m}{m^{2} - {(2 n)}^{2}}$

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 $b = 2 n$ .

$(3 m - 6 n) \frac{2 m}{(m + 2 n) (m - (2 n))}$

Multiply $2$ by $- 1$ .

$(3 m - 6 n) \frac{2 m}{(m + 2 n) (m - 2 n)}$

Step 4

Multiply $3 m - 6 n$ by $\frac{2 m}{(m + 2 n) (m - 2 n)}$ .

$\frac{(3 m - 6 n) (2 m)}{(m + 2 n) (m - 2 n)}$

Step 5

Simplify the numerator.

Factor $3$ out of $3 m - 6 n$ .

Factor $3$ out of $3 m$ .

$\frac{(3 (m) - 6 n) \cdot 2 m}{(m + 2 n) (m - 2 n)}$

Factor $3$ out of $- 6 n$ .

$\frac{(3 (m) + 3 (- 2 n)) \cdot 2 m}{(m + 2 n) (m - 2 n)}$

Factor $3$ out of $3 (m) + 3 (- 2 n)$ .

$\frac{3 (m - 2 n) \cdot 2 m}{(m + 2 n) (m - 2 n)}$

Multiply $2$ by $3$ .

$\frac{6 (m - 2 n) m}{(m + 2 n) (m - 2 n)}$

Step 6

Cancel the common factor of $m - 2 n$ .

Cancel the common factor.

$\frac{6 (m - 2 n) m}{(m + 2 n) (m - 2 n)}$

Rewrite the expression.

$\frac{6 m}{m + 2 n}$