Algebra Examples

$\frac{6 x^{3} y^{2} - 6 x^{3}}{3 x^{3} y^{3} - 3 x^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $6 x^{3}$ out of $6 x^{3} y^{2} - 6 x^{3}$ .

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Step 1.1.1

Factor $6 x^{3}$ out of $6 x^{3} y^{2}$ .

$\frac{6 x^{3} (y^{2}) - 6 x^{3}}{3 x^{3} y^{3} - 3 x^{3}}$

Step 1.1.2

Factor $6 x^{3}$ out of $- 6 x^{3}$ .

$\frac{6 x^{3} (y^{2}) + 6 x^{3} (- 1)}{3 x^{3} y^{3} - 3 x^{3}}$

Step 1.1.3

Factor $6 x^{3}$ out of $6 x^{3} (y^{2}) + 6 x^{3} (- 1)$ .

$\frac{6 x^{3} (y^{2} - 1)}{3 x^{3} y^{3} - 3 x^{3}}$

Step 1.2

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

$\frac{6 x^{3} (y^{2} - 1^{2})}{3 x^{3} y^{3} - 3 x^{3}}$

Step 1.3

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

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} y^{3} - 3 x^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $3 x^{3}$ out of $3 x^{3} y^{3} - 3 x^{3}$ .

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Step 2.1.1

Factor $3 x^{3}$ out of $3 x^{3} y^{3}$ .

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} (y^{3}) - 3 x^{3}}$

Step 2.1.2

Factor $3 x^{3}$ out of $- 3 x^{3}$ .

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} (y^{3}) + 3 x^{3} (- 1)}$

Step 2.1.3

Factor $3 x^{3}$ out of $3 x^{3} (y^{3}) + 3 x^{3} (- 1)$ .

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} (y^{3} - 1)}$

Step 2.2

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

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} (y^{3} - 1^{3})}$

Step 2.3

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 $b = 1$ .

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} ((y - 1) (y^{2} + y \cdot 1 + 1^{2}))}$

Step 2.4

Simplify.

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Step 2.4.1

Multiply $y$ by $1$ .

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} ((y - 1) (y^{2} + y + 1^{2}))}$

Step 2.4.2

One to any power is one.

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} ((y - 1) (y^{2} + y + 1))}$

$\frac{6 x^{3} (y + 1) (y - 1)}{3 x^{3} (y - 1) (y^{2} + y + 1)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $6$ and $3$ .

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Step 3.1.1

Factor $3$ out of $6 x^{3} (y + 1) (y - 1)$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $3$ out of $3 x^{3} (y - 1) (y^{2} + y + 1)$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x^{3}$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $y - 1$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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