Simplify (27y)/(y^2-1)*(y^2+y)/(36y^2)

$\frac{27 y}{y^{2} - 1} \cdot \frac{y^{2} + y}{36 y^{2}}$

Step 1

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$\frac{27 y}{y^{2} - 1^{2}} \cdot \frac{y^{2} + y}{36 y^{2}}$

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 $b = 1$ .

$\frac{27 y}{(y + 1) (y - 1)} \cdot \frac{y^{2} + y}{36 y^{2}}$

Step 2

Simplify terms.

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

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

$\frac{27 y}{(y + 1) (y - 1)} \cdot \frac{y \cdot y + y}{36 y^{2}}$

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

$\frac{27 y}{(y + 1) (y - 1)} \cdot \frac{y \cdot y + y^{1}}{36 y^{2}}$

Factor $y$ out of $y^{1}$ .

$\frac{27 y}{(y + 1) (y - 1)} \cdot \frac{y \cdot y + y \cdot 1}{36 y^{2}}$

Factor $y$ out of $y \cdot y + y \cdot 1$ .

$\frac{27 y}{(y + 1) (y - 1)} \cdot \frac{y (y + 1)}{36 y^{2}}$

Combine.

$\frac{27 y (y (y + 1))}{(y + 1) (y - 1) (36 y^{2})}$

Step 3

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

Move $y$ .

$\frac{27 (y \cdot y) (y + 1)}{(y + 1) (y - 1) (36 y^{2})}$

Multiply $y$ by $y$ .

$\frac{27 y^{2} (y + 1)}{(y + 1) (y - 1) (36 y^{2})}$

Step 4

Cancel the common factor of $27$ and $36$ .

Factor $9$ out of $27 y^{2} (y + 1)$ .

$\frac{9 (3 y^{2} (y + 1))}{(y + 1) (y - 1) (36 y^{2})}$

Cancel the common factors.

Factor $9$ out of $(y + 1) (y - 1) (36 y^{2})$ .

$\frac{9 (3 y^{2} (y + 1))}{9 ((y + 1) (y - 1) (4 y^{2}))}$

Cancel the common factor.

$\frac{9 (3 y^{2} (y + 1))}{9 ((y + 1) (y - 1) (4 y^{2}))}$

Rewrite the expression.

$\frac{3 y^{2} (y + 1)}{(y + 1) (y - 1) (4 y^{2})}$

Step 5

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

Cancel the common factor.

$\frac{3 y^{2} (y + 1)}{(y + 1) (y - 1) (4 y^{2})}$

Rewrite the expression.

$\frac{3 (y + 1)}{(y + 1) (y - 1) \cdot (4)}$

Step 6

Cancel the common factor of $y + 1$ .

Cancel the common factor.

$\frac{3 (y + 1)}{(y + 1) (y - 1) \cdot 4}$

Rewrite the expression.

$\frac{3}{(y - 1) \cdot (4)}$

Step 7

Move $4$ to the left of $y - 1$ .

$\frac{3}{4 (y - 1)}$

Simplify (27y)/(y^2-1)*(y^2+y)/(36y^2)

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