Algebra Examples

$\frac{a^{3} - b^{3}}{a^{3} b^{3} + a^{2} b^{4} + a b^{5}}$

Step 1

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 $a = a$ and $b = b$ .

$\frac{(a - b) (a^{2} + a b + b^{2})}{a^{3} b^{3} + a^{2} b^{4} + a b^{5}}$

Step 2

Simplify terms.

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Step 2.1

Factor $a b^{3}$ out of $a^{3} b^{3} + a^{2} b^{4} + a b^{5}$ .

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Step 2.1.1

Factor $a b^{3}$ out of $a^{3} b^{3}$ .

$\frac{(a - b) (a^{2} + a b + b^{2})}{a b^{3} (a^{2}) + a^{2} b^{4} + a b^{5}}$

Step 2.1.2

Factor $a b^{3}$ out of $a^{2} b^{4}$ .

$\frac{(a - b) (a^{2} + a b + b^{2})}{a b^{3} (a^{2}) + a b^{3} (a b) + a b^{5}}$

Step 2.1.3

Factor $a b^{3}$ out of $a b^{5}$ .

$\frac{(a - b) (a^{2} + a b + b^{2})}{a b^{3} (a^{2}) + a b^{3} (a b) + a b^{3} (b^{2})}$

Step 2.1.4

Factor $a b^{3}$ out of $a b^{3} (a^{2}) + a b^{3} (a b)$ .

$\frac{(a - b) (a^{2} + a b + b^{2})}{a b^{3} (a^{2} + a b) + a b^{3} (b^{2})}$

Step 2.1.5

Factor $a b^{3}$ out of $a b^{3} (a^{2} + a b) + a b^{3} (b^{2})$ .

$\frac{(a - b) (a^{2} + a b + b^{2})}{a b^{3} (a^{2} + a b + b^{2})}$

Step 2.2

Cancel the common factor of $a^{2} + a b + b^{2}$ .

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Step 2.2.1

Cancel the common factor.

$\frac{(a - b) (a^{2} + a b + b^{2})}{a b^{3} (a^{2} + a b + b^{2})}$

Step 2.2.2

Rewrite the expression.

$\frac{a - b}{a b^{3}}$