Simplify (45y)/(y^2-4)*(y^2+2y)/(18y^2)

$\frac{45 y}{y^{2} - 4} \cdot \frac{y^{2} + 2 y}{18 y^{2}}$

Step 1

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{45 y}{y^{2} - 2^{2}} \cdot \frac{y^{2} + 2 y}{18 y^{2}}$

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 $b = 2$ .

$\frac{45 y}{(y + 2) (y - 2)} \cdot \frac{y^{2} + 2 y}{18 y^{2}}$

Step 2

Simplify terms.

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

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

$\frac{45 y}{(y + 2) (y - 2)} \cdot \frac{y \cdot y + 2 y}{18 y^{2}}$

Factor $y$ out of $2 y$ .

$\frac{45 y}{(y + 2) (y - 2)} \cdot \frac{y \cdot y + y \cdot 2}{18 y^{2}}$

Factor $y$ out of $y \cdot y + y \cdot 2$ .

$\frac{45 y}{(y + 2) (y - 2)} \cdot \frac{y (y + 2)}{18 y^{2}}$

Combine.

$\frac{45 y (y (y + 2))}{(y + 2) (y - 2) (18 y^{2})}$

Step 3

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{45 (y \cdot y) (y + 2)}{(y + 2) (y - 2) (18 y^{2})}$

Multiply $y$ by $y$ .

$\frac{45 y^{2} (y + 2)}{(y + 2) (y - 2) (18 y^{2})}$

Step 4

Cancel the common factor of $45$ and $18$ .

Factor $9$ out of $45 y^{2} (y + 2)$ .

$\frac{9 (5 y^{2} (y + 2))}{(y + 2) (y - 2) (18 y^{2})}$

Cancel the common factors.

Factor $9$ out of $(y + 2) (y - 2) (18 y^{2})$ .

$\frac{9 (5 y^{2} (y + 2))}{9 ((y + 2) (y - 2) (2 y^{2}))}$

Cancel the common factor.

$\frac{9 (5 y^{2} (y + 2))}{9 ((y + 2) (y - 2) (2 y^{2}))}$

Rewrite the expression.

$\frac{5 y^{2} (y + 2)}{(y + 2) (y - 2) (2 y^{2})}$

Step 5

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

Cancel the common factor.

$\frac{5 y^{2} (y + 2)}{(y + 2) (y - 2) (2 y^{2})}$

Rewrite the expression.

$\frac{5 (y + 2)}{(y + 2) (y - 2) \cdot (2)}$

Step 6

Cancel the common factor of $y + 2$ .

Cancel the common factor.

$\frac{5 (y + 2)}{(y + 2) (y - 2) \cdot 2}$

Rewrite the expression.

$\frac{5}{(y - 2) \cdot (2)}$

Step 7

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

$\frac{5}{2 (y - 2)}$

Simplify (45y)/(y^2-4)*(y^2+2y)/(18y^2)

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