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.

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

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

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