Simplify (((x-y)^2)/(x+y))÷((3x+3y)/(x^2-y^2))

$\frac{{(x - y)}^{2}}{x + y} \div \frac{3 x + 3 y}{x^{2} - y^{2}}$

To divide by a fraction, multiply by its reciprocal.

$\frac{{(x - y)}^{2}}{x + y} \cdot \frac{x^{2} - y^{2}}{3 x + 3 y}$

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

$\frac{{(x - y)}^{2}}{x + y} \cdot \frac{(x + y) (x - y)}{3 x + 3 y}$

Simplify terms.

Factor $3$ out of $3 x + 3 y$ .

Factor $3$ out of $3 x$ .

$\frac{{(x - y)}^{2}}{x + y} \cdot \frac{(x + y) (x - y)}{3 (x) + 3 y}$

Factor $3$ out of $3 y$ .

$\frac{{(x - y)}^{2}}{x + y} \cdot \frac{(x + y) (x - y)}{3 (x) + 3 (y)}$

Factor $3$ out of $3 (x) + 3 (y)$ .

$\frac{{(x - y)}^{2}}{x + y} \cdot \frac{(x + y) (x - y)}{3 (x + y)}$

Cancel the common factor of $x + y$ .

Cancel the common factor.

$\frac{{(x - y)}^{2}}{x + y} \cdot \frac{(x + y) (x - y)}{3 (x + y)}$

Rewrite the expression.

${(x - y)}^{2} \frac{x - y}{3 (x + y)}$

Combine ${(x - y)}^{2}$ and $\frac{x - y}{3 (x + y)}$ .

$\frac{{(x - y)}^{2} (x - y)}{3 (x + y)}$

Multiply ${(x - y)}^{2}$ by $x - y$ by adding the exponents.

Multiply ${(x - y)}^{2}$ by $x - y$ .

Raise $x - y$ to the power of $1$ .

$\frac{{(x - y)}^{2} {(x - y)}^{1}}{3 (x + y)}$

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

$\frac{{(x - y)}^{2 + 1}}{3 (x + y)}$

Add $2$ and $1$ .

$\frac{{(x - y)}^{3}}{3 (x + y)}$