Algebra Examples

$\frac{y^{3} - 4 y}{16 y^{2}} \cdot \frac{- 8 y}{3 y^{3} - 12 y}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $y$ out of $y^{3} - 4 y$ .

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Step 1.1.1

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

$\frac{y \cdot y^{2} - 4 y}{16 y^{2}} \cdot \frac{- 8 y}{3 y^{3} - 12 y}$

Step 1.1.2

Factor $y$ out of $- 4 y$ .

$\frac{y \cdot y^{2} + y \cdot - 4}{16 y^{2}} \cdot \frac{- 8 y}{3 y^{3} - 12 y}$

Step 1.1.3

Factor $y$ out of $y \cdot y^{2} + y \cdot - 4$ .

$\frac{y (y^{2} - 4)}{16 y^{2}} \cdot \frac{- 8 y}{3 y^{3} - 12 y}$

Step 1.2

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

$\frac{y (y^{2} - 2^{2})}{16 y^{2}} \cdot \frac{- 8 y}{3 y^{3} - 12 y}$

Step 1.3

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 $b = 2$ .

$\frac{y (y + 2) (y - 2)}{16 y^{2}} \cdot \frac{- 8 y}{3 y^{3} - 12 y}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $3 y$ out of $3 y^{3} - 12 y$ .

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Step 2.1.1

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

$\frac{y (y + 2) (y - 2)}{16 y^{2}} \cdot \frac{- 8 y}{3 y (y^{2}) - 12 y}$

Step 2.1.2

Factor $3 y$ out of $- 12 y$ .

$\frac{y (y + 2) (y - 2)}{16 y^{2}} \cdot \frac{- 8 y}{3 y (y^{2}) + 3 y (- 4)}$

Step 2.1.3

Factor $3 y$ out of $3 y (y^{2}) + 3 y (- 4)$ .

$\frac{y (y + 2) (y - 2)}{16 y^{2}} \cdot \frac{- 8 y}{3 y (y^{2} - 4)}$

Step 2.2

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

$\frac{y (y + 2) (y - 2)}{16 y^{2}} \cdot \frac{- 8 y}{3 y (y^{2} - 2^{2})}$

Step 2.3

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 $b = 2$ .

$\frac{y (y + 2) (y - 2)}{16 y^{2}} \cdot \frac{- 8 y}{3 y (y + 2) (y - 2)}$

Step 3

Combine.

$\frac{y (y + 2) (y - 2) (- 8 y)}{16 y^{2} (3 y (y + 2) (y - 2))}$

Step 4

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

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Step 4.1

Move $y$ .

$\frac{y \cdot y (y + 2) (y - 2) \cdot - 8}{16 y^{2} (3 y (y + 2) (y - 2))}$

Step 4.2

Multiply $y$ by $y$ .

$\frac{y^{2} (y + 2) (y - 2) \cdot - 8}{16 y^{2} (3 y (y + 2) (y - 2))}$

Step 5

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 5.1

Move $y$ .

$\frac{y^{2} (y + 2) (y - 2) \cdot - 8}{16 (y \cdot y^{2}) ((3 (y + 2)) (y - 2))}$

Step 5.2

Multiply $y$ by $y^{2}$ .

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Step 5.2.1

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

$\frac{y^{2} (y + 2) (y - 2) \cdot - 8}{16 (y^{1} y^{2}) ((3 (y + 2)) (y - 2))}$

Step 5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{2} (y + 2) (y - 2) \cdot - 8}{16 y^{1 + 2} ((3 (y + 2)) (y - 2))}$

Step 5.3

Add $1$ and $2$ .

$\frac{y^{2} (y + 2) (y - 2) \cdot - 8}{16 y^{3} ((3 (y + 2)) (y - 2))}$

Step 6

Cancel the common factor of $y^{2}$ and $y^{3}$ .

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Step 6.1

Factor $y^{2}$ out of $y^{2} (y + 2) (y - 2) \cdot - 8$ .

$\frac{y^{2} (((y + 2) (y - 2)) \cdot - 8)}{16 y^{3} ((3 (y + 2)) (y - 2))}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $y^{2}$ out of $16 y^{3} ((3 (y + 2)) (y - 2))$ .

$\frac{y^{2} (((y + 2) (y - 2)) \cdot - 8)}{y^{2} (16 y ((3 (y + 2)) (y - 2)))}$

Step 6.2.2

Cancel the common factor.

$\frac{y^{2} (((y + 2) (y - 2)) \cdot - 8)}{y^{2} (16 y ((3 (y + 2)) (y - 2)))}$

Step 6.2.3

Rewrite the expression.

$\frac{((y + 2) (y - 2)) \cdot - 8}{16 y ((3 (y + 2)) (y - 2))}$

Step 7

Cancel the common factor of $y + 2$ .

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Step 7.1

Cancel the common factor.

$\frac{(y + 2) (y - 2) \cdot - 8}{16 y (3 (y + 2) (y - 2))}$

Step 7.2

Rewrite the expression.

$\frac{(y - 2) \cdot - 8}{16 y ((3) (y - 2))}$

Step 8

Cancel the common factor of $y - 2$ .

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Step 8.1

Cancel the common factor.

$\frac{(y - 2) \cdot - 8}{16 y (3 (y - 2))}$

Step 8.2

Rewrite the expression.

$\frac{- 8}{16 y \cdot (3)}$

Step 9

Cancel the common factor of $- 8$ and $16$ .

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Step 9.1

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{16 y \cdot (3)}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $8$ out of $16 y \cdot (3)$ .

$\frac{8 (- 1)}{8 (2 y \cdot 3)}$

Step 9.2.2

Cancel the common factor.

$\frac{8 \cdot - 1}{8 (2 y \cdot 3)}$

Step 9.2.3

Rewrite the expression.

$\frac{- 1}{2 y \cdot 3}$

Step 10

Simplify the expression.

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Step 10.1

Multiply $3$ by $2$ .

$\frac{- 1}{6 y}$

Step 10.2

Move the negative in front of the fraction.

$- \frac{1}{6 y}$