Basic Math Examples

$\frac{y^{3} + 729}{y^{2} - 81} \cdot \frac{y - 9}{9 y}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $729$ as $9^{3}$ .

$\frac{y^{3} + 9^{3}}{y^{2} - 81} \cdot \frac{y - 9}{9 y}$

Step 1.2

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 $b = 9$ .

$\frac{(y + 9) (y^{2} - y \cdot 9 + 9^{2})}{y^{2} - 81} \cdot \frac{y - 9}{9 y}$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $9$ by $- 1$ .

$\frac{(y + 9) (y^{2} - 9 y + 9^{2})}{y^{2} - 81} \cdot \frac{y - 9}{9 y}$

Step 1.3.2

Raise $9$ to the power of $2$ .

$\frac{(y + 9) (y^{2} - 9 y + 81)}{y^{2} - 81} \cdot \frac{y - 9}{9 y}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $81$ as $9^{2}$ .

$\frac{(y + 9) (y^{2} - 9 y + 81)}{y^{2} - 9^{2}} \cdot \frac{y - 9}{9 y}$

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 $b = 9$ .

$\frac{(y + 9) (y^{2} - 9 y + 81)}{(y + 9) (y - 9)} \cdot \frac{y - 9}{9 y}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $y - 9$ .

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Step 3.1.1

Factor $y - 9$ out of $(y + 9) (y - 9)$ .

$\frac{(y + 9) (y^{2} - 9 y + 81)}{(y - 9) (y + 9)} \cdot \frac{y - 9}{9 y}$

Step 3.1.2

Cancel the common factor.

$\frac{(y + 9) (y^{2} - 9 y + 81)}{(y - 9) (y + 9)} \cdot \frac{y - 9}{9 y}$

Step 3.1.3

Rewrite the expression.

$\frac{(y + 9) (y^{2} - 9 y + 81)}{y + 9} \cdot \frac{1}{9 y}$

Step 3.2

Multiply $\frac{(y + 9) (y^{2} - 9 y + 81)}{y + 9}$ by $\frac{1}{9 y}$ .

$\frac{(y + 9) (y^{2} - 9 y + 81)}{(y + 9) (9 y)}$

Step 3.3

Cancel the common factor of $y + 9$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(y + 9) (y^{2} - 9 y + 81)}{(y + 9) (9 y)}$

Step 3.3.2

Rewrite the expression.

$\frac{y^{2} - 9 y + 81}{9 y}$