Precalculus Examples

$\frac{\frac{8}{y} - \frac{y}{8}}{y + 8}$

Step 1

Multiply the numerator and denominator of the fraction by $y \cdot 8$ .

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Step 1.1

Multiply $\frac{\frac{8}{y} - \frac{y}{8}}{y + 8}$ by $\frac{y \cdot 8}{y \cdot 8}$ .

$\frac{y \cdot 8}{y \cdot 8} \cdot \frac{\frac{8}{y} - \frac{y}{8}}{y + 8}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

Factor $y$ out of $y \cdot 8$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $8$ by $8$ .

$\frac{64 + y \cdot 8 (- \frac{y}{8})}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 3.3

Cancel the common factor of $8$ .

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Step 3.3.1

Move the leading negative in $- \frac{y}{8}$ into the numerator.

$\frac{64 + y \cdot 8 \frac{- y}{8}}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 3.3.2

Factor $8$ out of $y \cdot 8$ .

$\frac{64 + 8 \cdot y \frac{- y}{8}}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 3.3.3

Cancel the common factor.

$\frac{64 + 8 \cdot y \frac{- y}{8}}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 3.3.4

Rewrite the expression.

$\frac{64 + y (- y)}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 3.4

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

$\frac{64 - (y^{1} y)}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 3.5

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

$\frac{64 - (y^{1} y^{1})}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 3.6

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

$\frac{64 - y^{1 + 1}}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 3.7

Add $1$ and $1$ .

$\frac{64 - y^{2}}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $64$ as $8^{2}$ .

$\frac{8^{2} - y^{2}}{y \cdot 8 y + y \cdot 8 \cdot 8}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = y$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

Move $y$ .

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

Step 5.1.2

Multiply $y$ by $y$ .

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

Step 5.2

Move $8$ to the left of $y^{2}$ .

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

Step 5.3

Move $8$ to the left of $y$ .

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

Step 5.4

Multiply $8$ by $8$ .

$\frac{(8 + y) (8 - y)}{8 y^{2} + 64 y}$

Step 5.5

Factor $8 y$ out of $8 y^{2} + 64 y$ .

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Step 5.5.1

Factor $8 y$ out of $8 y^{2}$ .

$\frac{(8 + y) (8 - y)}{8 y (y) + 64 y}$

Step 5.5.2

Factor $8 y$ out of $64 y$ .

$\frac{(8 + y) (8 - y)}{8 y (y) + 8 y (8)}$

Step 5.5.3

Factor $8 y$ out of $8 y (y) + 8 y (8)$ .

$\frac{(8 + y) (8 - y)}{8 y (y + 8)}$

Step 6

Cancel the common factor of $8 + y$ and $y + 8$ .

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Step 6.1

Reorder terms.

$\frac{(y + 8) (8 - y)}{8 y (y + 8)}$

Step 6.2

Cancel the common factor.

$\frac{(y + 8) (8 - y)}{8 y (y + 8)}$

Step 6.3

Rewrite the expression.

$\frac{8 - y}{8 y}$