Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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 $b = 1$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $1$ as $1^{3}$ .

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

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 $b = 1$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Multiply $y$ by $1$ .

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

Step 2.3.2

One to any power is one.

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $y + 1$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y - 1$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

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

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