Basic Math Examples

$\frac{y^{3} - 27}{y^{3} + 27} \div \frac{y^{2} - 9}{y^{2} - 3 y + 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{3} - 27}{y^{3} + 27} \cdot \frac{y^{2} - 3 y + 9}{y^{2} - 9}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $27$ as $3^{3}$ .

$\frac{y^{3} - 3^{3}}{y^{3} + 27} \cdot \frac{y^{2} - 3 y + 9}{y^{2} - 9}$

Step 2.2

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 $b = 3$ .

$\frac{(y - 3) (y^{2} + y \cdot 3 + 3^{2})}{y^{3} + 27} \cdot \frac{y^{2} - 3 y + 9}{y^{2} - 9}$

Step 2.3

Simplify.

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Step 2.3.1

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

$\frac{(y - 3) (y^{2} + 3 \cdot y + 3^{2})}{y^{3} + 27} \cdot \frac{y^{2} - 3 y + 9}{y^{2} - 9}$

Step 2.3.2

Raise $3$ to the power of $2$ .

$\frac{(y - 3) (y^{2} + 3 y + 9)}{y^{3} + 27} \cdot \frac{y^{2} - 3 y + 9}{y^{2} - 9}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $27$ as $3^{3}$ .

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

Step 3.2

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

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

Step 3.3

Simplify.

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Step 3.3.1

Multiply $3$ by $- 1$ .

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

Step 3.3.2

Raise $3$ to the power of $2$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $y^{2} - 3 y + 9$ .

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Step 4.1.1

Factor $y^{2} - 3 y + 9$ out of $(y + 3) (y^{2} - 3 y + 9)$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

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

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

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

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

Step 5.3

Combine exponents.

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Step 5.3.1

Raise $y + 3$ to the power of $1$ .

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

Step 5.3.2

Raise $y + 3$ to the power of $1$ .

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

Step 5.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3.4

Add $1$ and $1$ .

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

Step 6

Cancel the common factor of $y - 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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