Simplify ((2ab)/(a^2-b^2))÷((4ab)/((a-b)^2))

$\frac{2 a b}{a^{2} - b^{2}} \div \frac{4 a b}{{(a - b)}^{2}}$

To divide by a fraction, multiply by its reciprocal.

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

Simplify terms.

Cancel the common factor of $2 a b$ .

Factor $2 a b$ out of $4 a b$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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 $b = b$ .

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of ${(a - b)}^{2}$ and $a - b$ .

Factor $a - b$ out of ${(a - b)}^{2}$ .

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

Cancel the common factors.

Factor $a - b$ out of $(a + b) (a - b) \cdot 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move $2$ to the left of $a + b$ .

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