Pre-Algebra Examples

$\frac{y^{2} - 49}{y} \div \frac{y - 7}{y + 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} - 49}{y} \cdot \frac{y + 1}{y - 7}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

$\frac{y^{2} - 7^{2}}{y} \cdot \frac{y + 1}{y - 7}$

Step 2.2

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 $b = 7$ .

$\frac{(y + 7) (y - 7)}{y} \cdot \frac{y + 1}{y - 7}$

Step 3

Cancel the common factor of $y - 7$ .

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Step 3.1

Factor $y - 7$ out of $(y + 7) (y - 7)$ .

$\frac{(y - 7) (y + 7)}{y} \cdot \frac{y + 1}{y - 7}$

Step 3.2

Cancel the common factor.

$\frac{(y - 7) (y + 7)}{y} \cdot \frac{y + 1}{y - 7}$

Step 3.3

Rewrite the expression.

$\frac{y + 7}{y} (y + 1)$

Step 4

Multiply $\frac{y + 7}{y}$ by $y + 1$ .

$\frac{(y + 7) (y + 1)}{y}$

Step 5

Expand $(y + 7) (y + 1)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{y (y + 1) + 7 (y + 1)}{y}$

Step 5.2

Apply the distributive property.

$\frac{y \cdot y + y \cdot 1 + 7 (y + 1)}{y}$

Step 5.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot 1 + 7 y + 7 \cdot 1}{y}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $y$ by $y$ .

$\frac{y^{2} + y \cdot 1 + 7 y + 7 \cdot 1}{y}$

Step 6.1.2

Multiply $y$ by $1$ .

$\frac{y^{2} + y + 7 y + 7 \cdot 1}{y}$

Step 6.1.3

Multiply $7$ by $1$ .

$\frac{y^{2} + y + 7 y + 7}{y}$

Step 6.2

Add $y$ and $7 y$ .

$\frac{y^{2} + 8 y + 7}{y}$

Step 7

Split the fraction $\frac{y^{2} + 8 y + 7}{y}$ into two fractions.

$\frac{y^{2} + 8 y}{y} + \frac{7}{y}$

Step 8

Split the fraction $\frac{y^{2} + 8 y}{y}$ into two fractions.

$\frac{y^{2}}{y} + \frac{8 y}{y} + \frac{7}{y}$

Step 9

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

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Step 9.1

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

$\frac{y \cdot y}{y} + \frac{8 y}{y} + \frac{7}{y}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

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

$\frac{y \cdot y}{y^{1}} + \frac{8 y}{y} + \frac{7}{y}$

Step 9.2.2

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

$\frac{y \cdot y}{y \cdot 1} + \frac{8 y}{y} + \frac{7}{y}$

Step 9.2.3

Cancel the common factor.

$\frac{y \cdot y}{y \cdot 1} + \frac{8 y}{y} + \frac{7}{y}$

Step 9.2.4

Rewrite the expression.

$\frac{y}{1} + \frac{8 y}{y} + \frac{7}{y}$

Step 9.2.5

Divide $y$ by $1$ .

$y + \frac{8 y}{y} + \frac{7}{y}$

Step 10

Cancel the common factor of $y$ .

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Step 10.1

Cancel the common factor.

$y + \frac{8 y}{y} + \frac{7}{y}$

Step 10.2

Divide $8$ by $1$ .

$y + 8 + \frac{7}{y}$