Simplify ((y^2-9)/(3y))/((y^2+3y)/(2y^3))

$\frac{\frac{y^{2} - 9}{3 y}}{\frac{y^{2} + 3 y}{2 y^{3}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - 9}{3 y} \cdot \frac{2 y^{3}}{y^{2} + 3 y}$

Step 2

Simplify terms.

Cancel the common factor of $y$ .

Factor $y$ out of $3 y$ .

$\frac{y^{2} - 9}{y \cdot 3} \cdot \frac{2 y^{3}}{y^{2} + 3 y}$

Factor $y$ out of $2 y^{3}$ .

$\frac{y^{2} - 9}{y \cdot 3} \cdot \frac{y (2 y^{2})}{y^{2} + 3 y}$

Cancel the common factor.

$\frac{y^{2} - 9}{y \cdot 3} \cdot \frac{y (2 y^{2})}{y^{2} + 3 y}$

Rewrite the expression.

$\frac{y^{2} - 9}{3} \cdot \frac{2 y^{2}}{y^{2} + 3 y}$

Multiply $\frac{y^{2} - 9}{3}$ by $\frac{2 y^{2}}{y^{2} + 3 y}$ .

$\frac{(y^{2} - 9) (2 y^{2})}{3 (y^{2} + 3 y)}$

Step 3

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 3$ .

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

Step 4

Factor $y$ out of $y^{2} + 3 y$ .

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

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

Factor $y$ out of $3 y$ .

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

Factor $y$ out of $y \cdot y + y \cdot 3$ .

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

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

Step 5

Cancel the common factor of $y + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Cancel the common factor of $y^{2}$ and $y$ .

Factor $y$ out of $(y - 3) \cdot 2 y^{2}$ .

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

Cancel the common factors.

Factor $y$ out of $3 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Simplify the expression.

Move $2$ to the left of $y - 3$ .

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

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

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

Simplify ((y^2-9)/(3y))/((y^2+3y)/(2y^3))

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