Algebra Examples

$x^{15} - 64$

Step 1

Rewrite $x^{15}$ as ${(x^{5})}^{3}$ .

${(x^{5})}^{3} - 64$

Step 2

Rewrite $64$ as $4^{3}$ .

${(x^{5})}^{3} - 4^{3}$

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 $a = x^{5}$ and $b = 4$ .

$(x^{5} - 4) ({(x^{5})}^{2} + x^{5} \cdot 4 + 4^{2})$

Step 4

Simplify.

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Step 4.1

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 4.1.1

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

$(x^{5} - 4) (x^{5 \cdot 2} + x^{5} \cdot 4 + 4^{2})$

Step 4.1.2

Multiply $5$ by $2$ .

$(x^{5} - 4) (x^{10} + x^{5} \cdot 4 + 4^{2})$

Step 4.2

Move $4$ to the left of $x^{5}$ .

$(x^{5} - 4) (x^{10} + 4 x^{5} + 4^{2})$

Step 4.3

Raise $4$ to the power of $2$ .

$(x^{5} - 4) (x^{10} + 4 x^{5} + 16)$