Algebra Examples

$a^{6} - b^{6}$

Step 1

Rewrite $a^{6}$ as ${(a^{2})}^{3}$ .

${(a^{2})}^{3} - b^{6}$

Step 2

Rewrite $b^{6}$ as ${(b^{2})}^{3}$ .

${(a^{2})}^{3} - {(b^{2})}^{3}$

Step 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 $a = a^{2}$ and $b = b^{2}$ .

$(a^{2} - b^{2}) ({(a^{2})}^{2} + a^{2} b^{2} + {(b^{2})}^{2})$

Step 4

Simplify.

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Step 4.1

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

$(a + b) (a - b) ({(a^{2})}^{2} + a^{2} b^{2} + {(b^{2})}^{2})$

Step 4.2

Multiply the exponents in ${(a^{2})}^{2}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.2

Multiply $2$ by $2$ .

$(a + b) (a - b) (a^{4} + a^{2} b^{2} + {(b^{2})}^{2})$

Step 4.3

Multiply the exponents in ${(b^{2})}^{2}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(a + b) (a - b) (a^{4} + a^{2} b^{2} + b^{2 \cdot 2})$

Step 4.3.2

Multiply $2$ by $2$ .

$(a + b) (a - b) (a^{4} + a^{2} b^{2} + b^{4})$

Step 4.4

Factor.

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Step 4.4.1

Rewrite $a^{4} + a^{2} b^{2} + b^{4}$ in a factored form.

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Step 4.4.1.1

Rewrite the middle term.

$(a + b) (a - b) (a^{4} + 2 a^{2} b^{2} - a^{2} b^{2} + b^{4})$

Step 4.4.1.2

Rearrange terms.

$(a + b) (a - b) (a^{4} + 2 a^{2} b^{2} + b^{4} - a^{2} b^{2})$

Step 4.4.1.3

Factor first three terms by perfect square rule.

$(a + b) (a - b) ({(a^{2} + b^{2})}^{2} - a^{2} b^{2})$

Step 4.4.1.4

Rewrite $a^{2} b^{2}$ as ${(a b)}^{2}$ .

$(a + b) (a - b) ({(a^{2} + b^{2})}^{2} - {(a b)}^{2})$

Step 4.4.1.5

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

$(a + b) (a - b) ((a^{2} + b^{2} + a b) (a^{2} + b^{2} - (a b)))$

Step 4.4.1.6

Remove parentheses.

$(a + b) (a - b) ((a^{2} + b^{2} + a b) (a^{2} + b^{2} - a b))$

Step 4.4.2

Remove unnecessary parentheses.

$(a + b) (a - b) (a^{2} + b^{2} + a b) (a^{2} + b^{2} - a b)$