Pre-Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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 $b = 8$ .

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

Step 2

To write $\frac{(y + 8) (y - 8)}{y}$ as a fraction with a common denominator, multiply by $\frac{y + 8}{y + 8}$ .

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

Step 3

To write $\frac{y - 8}{y + 8}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 4

Write each expression with a common denominator of $y (y + 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(y + 8) y$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.2

Expand $(y + 8) (y + 8)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $y$ by $y$ .

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

Step 6.3.1.2

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

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

Step 6.3.1.3

Multiply $8$ by $8$ .

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

Step 6.3.2

Add $8 y$ and $8 y$ .

$\frac{(y - 8) (y^{2} + 16 y + 64 + y)}{y (y + 8)}$

Step 6.4

Add $16 y$ and $y$ .

$\frac{(y - 8) (y^{2} + 17 y + 64)}{y (y + 8)}$