Pre-Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $y^{2}$ .

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Step 1.1

Multiply $\frac{1 + \frac{1}{y}}{1 - \frac{1}{y^{2}}}$ by $\frac{y^{2}}{y^{2}}$ .

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

Step 1.2

Combine.

$\frac{y^{2} (1 + \frac{1}{y})}{y^{2} (1 - \frac{1}{y^{2}})}$

Step 2

Apply the distributive property.

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

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 $y^{2}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

Move the leading negative in $- \frac{1}{y^{2}}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $y$ out of $y^{2} \cdot 1 + y$ .

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Step 4.1.1

Factor $y$ out of $y^{2} \cdot 1$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Factor $y$ out of $y (y \cdot 1) + y \cdot 1$ .

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

Step 4.2

Multiply $y$ by $1$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $y^{2} \cdot 1$ as ${(y \cdot 1)}^{2}$ .

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

Step 5.2

Rewrite $1$ as $1^{2}$ .

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

Step 5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y \cdot 1$ and $b = 1$ .

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

Step 5.4

Simplify.

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Step 5.4.1

Multiply $y$ by $1$ .

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

Step 5.4.2

Multiply $y$ by $1$ .

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

Step 6

Cancel the common factor of $y + 1$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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