Basic Math Examples

$\frac{9 - \frac{1}{y^{2}}}{3 - \frac{1}{y}}$

Step 1

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

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Step 1.1

Multiply $\frac{9 - \frac{1}{y^{2}}}{3 - \frac{1}{y}}$ by $\frac{y^{2}}{y^{2}}$ .

$\frac{y^{2}}{y^{2}} \cdot \frac{9 - \frac{1}{y^{2}}}{3 - \frac{1}{y}}$

Step 1.2

Combine.

$\frac{y^{2} (9 - \frac{1}{y^{2}})}{y^{2} (3 - \frac{1}{y})}$

Step 2

Apply the distributive property.

$\frac{y^{2} \cdot 9 + y^{2} (- \frac{1}{y^{2}})}{y^{2} \cdot 3 + y^{2} (- \frac{1}{y})}$

Step 3

Simplify by cancelling.

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Step 3.1

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

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Step 3.1.1

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

$\frac{y^{2} \cdot 9 + y^{2} \frac{- 1}{y^{2}}}{y^{2} \cdot 3 + y^{2} (- \frac{1}{y})}$

Step 3.1.2

Cancel the common factor.

$\frac{y^{2} \cdot 9 + y^{2} \frac{- 1}{y^{2}}}{y^{2} \cdot 3 + y^{2} (- \frac{1}{y})}$

Step 3.1.3

Rewrite the expression.

$\frac{y^{2} \cdot 9 - 1}{y^{2} \cdot 3 + y^{2} (- \frac{1}{y})}$

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

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

$\frac{y^{2} \cdot 9 - 1}{y^{2} \cdot 3 + y^{2} \frac{- 1}{y}}$

Step 3.2.2

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

$\frac{y^{2} \cdot 9 - 1}{y^{2} \cdot 3 + y \cdot y \frac{- 1}{y}}$

Step 3.2.3

Cancel the common factor.

$\frac{y^{2} \cdot 9 - 1}{y^{2} \cdot 3 + y \cdot y \frac{- 1}{y}}$

Step 3.2.4

Rewrite the expression.

$\frac{y^{2} \cdot 9 - 1}{y^{2} \cdot 3 + y \cdot - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $y^{2} \cdot 9$ as ${(y \cdot 3)}^{2}$ .

$\frac{{(y \cdot 3)}^{2} - 1}{y^{2} \cdot 3 + y \cdot - 1}$

Step 4.2

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

$\frac{{(y \cdot 3)}^{2} - 1^{2}}{y^{2} \cdot 3 + y \cdot - 1}$

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y \cdot 3$ and $b = 1$ .

$\frac{(y \cdot 3 + 1) (y \cdot 3 - 1)}{y^{2} \cdot 3 + y \cdot - 1}$

Step 4.4

Simplify.

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Step 4.4.1

Move $3$ to the left of $y$ .

$\frac{(3 \cdot y + 1) (y \cdot 3 - 1)}{y^{2} \cdot 3 + y \cdot - 1}$

Step 4.4.2

Move $3$ to the left of $y$ .

$\frac{(3 y + 1) (3 y - 1)}{y^{2} \cdot 3 + y \cdot - 1}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $y$ out of $y^{2} \cdot 3 + y \cdot - 1$ .

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Step 5.1.1

Factor $y$ out of $y^{2} \cdot 3$ .

$\frac{(3 y + 1) (3 y - 1)}{y (y \cdot 3) + y \cdot - 1}$

Step 5.1.2

Factor $y$ out of $y \cdot - 1$ .

$\frac{(3 y + 1) (3 y - 1)}{y (y \cdot 3) + y (- 1)}$

Step 5.1.3

Factor $y$ out of $y (y \cdot 3) + y (- 1)$ .

$\frac{(3 y + 1) (3 y - 1)}{y (y \cdot 3 - 1)}$

Step 5.2

Move $3$ to the left of $y$ .

$\frac{(3 y + 1) (3 y - 1)}{y (3 y - 1)}$

Step 6

Cancel the common factor of $3 y - 1$ .

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Step 6.1

Cancel the common factor.

$\frac{(3 y + 1) (3 y - 1)}{y (3 y - 1)}$

Step 6.2

Rewrite the expression.

$\frac{3 y + 1}{y}$