Pre-Algebra Examples

$\frac{\frac{y^{2} - 25}{y}}{\frac{y - 5}{y + 7}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - 25}{y} \cdot \frac{y + 7}{y - 5}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $25$ as $5^{2}$ .

$\frac{y^{2} - 5^{2}}{y} \cdot \frac{y + 7}{y - 5}$

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 $b = 5$ .

$\frac{(y + 5) (y - 5)}{y} \cdot \frac{y + 7}{y - 5}$

Step 3

Cancel the common factor of $y - 5$ .

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Step 3.1

Factor $y - 5$ out of $(y + 5) (y - 5)$ .

$\frac{(y - 5) (y + 5)}{y} \cdot \frac{y + 7}{y - 5}$

Step 3.2

Cancel the common factor.

$\frac{(y - 5) (y + 5)}{y} \cdot \frac{y + 7}{y - 5}$

Step 3.3

Rewrite the expression.

$\frac{y + 5}{y} (y + 7)$

Step 4

Multiply $\frac{y + 5}{y}$ by $y + 7$ .

$\frac{(y + 5) (y + 7)}{y}$

Step 5

Expand $(y + 5) (y + 7)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{y (y + 7) + 5 (y + 7)}{y}$

Step 5.2

Apply the distributive property.

$\frac{y \cdot y + y \cdot 7 + 5 (y + 7)}{y}$

Step 5.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot 7 + 5 y + 5 \cdot 7}{y}$

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 $y$ .

$\frac{y^{2} + y \cdot 7 + 5 y + 5 \cdot 7}{y}$

Step 6.1.2

Move $7$ to the left of $y$ .

$\frac{y^{2} + 7 \cdot y + 5 y + 5 \cdot 7}{y}$

Step 6.1.3

Multiply $5$ by $7$ .

$\frac{y^{2} + 7 y + 5 y + 35}{y}$

Step 6.2

Add $7 y$ and $5 y$ .

$\frac{y^{2} + 12 y + 35}{y}$

Step 7

Split the fraction $\frac{y^{2} + 12 y + 35}{y}$ into two fractions.

$\frac{y^{2} + 12 y}{y} + \frac{35}{y}$

Step 8

Split the fraction $\frac{y^{2} + 12 y}{y}$ into two fractions.

$\frac{y^{2}}{y} + \frac{12 y}{y} + \frac{35}{y}$

Step 9

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

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Step 9.1

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

$\frac{y \cdot y}{y} + \frac{12 y}{y} + \frac{35}{y}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

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

$\frac{y \cdot y}{y^{1}} + \frac{12 y}{y} + \frac{35}{y}$

Step 9.2.2

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

$\frac{y \cdot y}{y \cdot 1} + \frac{12 y}{y} + \frac{35}{y}$

Step 9.2.3

Cancel the common factor.

$\frac{y \cdot y}{y \cdot 1} + \frac{12 y}{y} + \frac{35}{y}$

Step 9.2.4

Rewrite the expression.

$\frac{y}{1} + \frac{12 y}{y} + \frac{35}{y}$

Step 9.2.5

Divide $y$ by $1$ .

$y + \frac{12 y}{y} + \frac{35}{y}$

Step 10

Cancel the common factor of $y$ .

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Step 10.1

Cancel the common factor.

$y + \frac{12 y}{y} + \frac{35}{y}$

Step 10.2

Divide $12$ by $1$ .

$y + 12 + \frac{35}{y}$