Precalculus Examples

$a^{12} - b^{3}$

Step 1

Rewrite $a^{12}$ as ${(a^{4})}^{3}$ .

${(a^{4})}^{3} - b^{3}$

Step 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 $a = a^{4}$ and $b = b$ .

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

Step 3

Multiply the exponents in ${(a^{4})}^{2}$ .

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Step 3.1

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

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

Step 3.2

Multiply $4$ by $2$ .

$(a^{4} - b) (a^{8} + a^{4} b + b^{2})$