Basic Math Examples

$\frac{y^{3} - 8}{y^{2} + 2 y + 4} \div (y^{2} - 4)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{y^{3} - 8}{y^{2} + 2 y + 4}}{y^{2} - 4}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{3} - 8}{y^{2} + 2 y + 4} \cdot \frac{1}{y^{2} - 4}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $8$ as $2^{3}$ .

$\frac{y^{3} - 2^{3}}{y^{2} + 2 y + 4} \cdot \frac{1}{y^{2} - 4}$

Step 3.2

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

$\frac{(y - 2) (y^{2} + y \cdot 2 + 2^{2})}{y^{2} + 2 y + 4} \cdot \frac{1}{y^{2} - 4}$

Step 3.3

Simplify.

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Step 3.3.1

Move $2$ to the left of $y$ .

$\frac{(y - 2) (y^{2} + 2 \cdot y + 2^{2})}{y^{2} + 2 y + 4} \cdot \frac{1}{y^{2} - 4}$

Step 3.3.2

Raise $2$ to the power of $2$ .

$\frac{(y - 2) (y^{2} + 2 y + 4)}{y^{2} + 2 y + 4} \cdot \frac{1}{y^{2} - 4}$

Step 4

Cancel the common factor of $y^{2} + 2 y + 4$ .

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Step 4.1

Cancel the common factor.

$\frac{(y - 2) (y^{2} + 2 y + 4)}{y^{2} + 2 y + 4} \cdot \frac{1}{y^{2} - 4}$

Step 4.2

Divide $y - 2$ by $1$ .

$(y - 2) \frac{1}{y^{2} - 4}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $4$ as $2^{2}$ .

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

Step 5.2

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 $b = 2$ .

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

Step 6

Cancel the common factor of $y - 2$ .

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Step 6.1

Factor $y - 2$ out of $(y + 2) (y - 2)$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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