Algebra Examples

$\frac{\frac{y^{2} - 36}{6 y}}{\frac{y^{2} + 6 y}{5 y^{3}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - 36}{6 y} \cdot \frac{5 y^{3}}{y^{2} + 6 y}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $y$ .

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Step 2.1.1

Factor $y$ out of $6 y$ .

$\frac{y^{2} - 36}{y \cdot 6} \cdot \frac{5 y^{3}}{y^{2} + 6 y}$

Step 2.1.2

Factor $y$ out of $5 y^{3}$ .

$\frac{y^{2} - 36}{y \cdot 6} \cdot \frac{y (5 y^{2})}{y^{2} + 6 y}$

Step 2.1.3

Cancel the common factor.

$\frac{y^{2} - 36}{y \cdot 6} \cdot \frac{y (5 y^{2})}{y^{2} + 6 y}$

Step 2.1.4

Rewrite the expression.

$\frac{y^{2} - 36}{6} \cdot \frac{5 y^{2}}{y^{2} + 6 y}$

Step 2.2

Multiply $\frac{y^{2} - 36}{6}$ by $\frac{5 y^{2}}{y^{2} + 6 y}$ .

$\frac{(y^{2} - 36) (5 y^{2})}{6 (y^{2} + 6 y)}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $36$ as $6^{2}$ .

$\frac{(y^{2} - 6^{2}) \cdot 5 y^{2}}{6 (y^{2} + 6 y)}$

Step 3.2

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 $b = 6$ .

$\frac{(y + 6) (y - 6) \cdot 5 y^{2}}{6 (y^{2} + 6 y)}$

Step 4

Factor $y$ out of $y^{2} + 6 y$ .

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Step 4.1

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

$\frac{(y + 6) (y - 6) \cdot 5 y^{2}}{6 (y \cdot y + 6 y)}$

Step 4.2

Factor $y$ out of $6 y$ .

$\frac{(y + 6) (y - 6) \cdot 5 y^{2}}{6 (y \cdot y + y \cdot 6)}$

Step 4.3

Factor $y$ out of $y \cdot y + y \cdot 6$ .

$\frac{(y + 6) (y - 6) \cdot 5 y^{2}}{6 (y (y + 6))}$

$\frac{(y + 6) (y - 6) \cdot 5 y^{2}}{6 y (y + 6)}$

Step 5

Cancel the common factor of $y + 6$ .

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Step 5.1

Cancel the common factor.

$\frac{(y + 6) (y - 6) \cdot 5 y^{2}}{6 y (y + 6)}$

Step 5.2

Rewrite the expression.

$\frac{(y - 6) \cdot 5 y^{2}}{6 y}$

Step 6

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

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Step 6.1

Factor $y$ out of $(y - 6) \cdot 5 y^{2}$ .

$\frac{y ((y - 6) \cdot 5 y)}{6 y}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $y$ out of $6 y$ .

$\frac{y ((y - 6) \cdot 5 y)}{y \cdot 6}$

Step 6.2.2

Cancel the common factor.

$\frac{y ((y - 6) \cdot 5 y)}{y \cdot 6}$

Step 6.2.3

Rewrite the expression.

$\frac{(y - 6) \cdot 5 y}{6}$

Step 7

Simplify the expression.

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Step 7.1

Move $5$ to the left of $y - 6$ .

$\frac{5 \cdot (y - 6) y}{6}$

Step 7.2

Reorder factors in $\frac{5 (y - 6) y}{6}$ .

$\frac{5 y (y - 6)}{6}$