Simplify ((x^2-9y^2)/(xy))/((2x+6y)/(x^2))

$\frac{\frac{x^{2} - 9 y^{2}}{x y}}{\frac{2 x + 6 y}{x^{2}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 9 y^{2}}{x y} \cdot \frac{x^{2}}{2 x + 6 y}$

Step 2

Simplify terms.

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

$\frac{x^{2} - 9 y^{2}}{x (y)} \cdot \frac{x^{2}}{2 x + 6 y}$

Factor $x$ out of $x^{2}$ .

$\frac{x^{2} - 9 y^{2}}{x (y)} \cdot \frac{x \cdot x}{2 x + 6 y}$

Cancel the common factor.

$\frac{x^{2} - 9 y^{2}}{x y} \cdot \frac{x \cdot x}{2 x + 6 y}$

Rewrite the expression.

$\frac{x^{2} - 9 y^{2}}{y} \cdot \frac{x}{2 x + 6 y}$

Multiply $\frac{x^{2} - 9 y^{2}}{y}$ by $\frac{x}{2 x + 6 y}$ .

$\frac{(x^{2} - 9 y^{2}) x}{y (2 x + 6 y)}$

Step 3

Simplify the numerator.

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

$\frac{(x^{2} - {(3 y)}^{2}) x}{y (2 x + 6 y)}$

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 $b = 3 y$ .

$\frac{(x + 3 y) (x - (3 y)) x}{y (2 x + 6 y)}$

Multiply $3$ by $- 1$ .

$\frac{(x + 3 y) (x - 3 y) x}{y (2 x + 6 y)}$

Step 4

Factor $2$ out of $2 x + 6 y$ .

Factor $2$ out of $2 x$ .

$\frac{(x + 3 y) (x - 3 y) x}{y (2 (x) + 6 y)}$

Factor $2$ out of $6 y$ .

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

Factor $2$ out of $2 (x) + 2 (3 y)$ .

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

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

Step 5

Cancel the common factor of $x + 3 y$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Simplify the expression.

Move $2$ to the left of $y$ .

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

Reorder factors in $\frac{(x - 3 y) x}{2 y}$ .

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

Simplify ((x^2-9y^2)/(xy))/((2x+6y)/(x^2))

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