Factor x^6-64=0

$x^{6} - 64 = 0$

Step 1

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

${(x^{2})}^{3} - 64 = 0$

Step 2

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

${(x^{2})}^{3} - 4^{3} = 0$

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 $a = x^{2}$ and $b = 4$ .

$(x^{2} - 4) ({(x^{2})}^{2} + x^{2} \cdot 4 + 4^{2}) = 0$

Step 4

Simplify.

Rewrite $4$ as $2^{2}$ .

$(x^{2} - 2^{2}) ({(x^{2})}^{2} + x^{2} \cdot 4 + 4^{2}) = 0$

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

$(x + 2) (x - 2) ({(x^{2})}^{2} + x^{2} \cdot 4 + 4^{2}) = 0$

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

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

$(x + 2) (x - 2) (x^{2 \cdot 2} + x^{2} \cdot 4 + 4^{2}) = 0$

Multiply $2$ by $2$ .

$(x + 2) (x - 2) (x^{4} + x^{2} \cdot 4 + 4^{2}) = 0$

Move $4$ to the left of $x^{2}$ .

$(x + 2) (x - 2) (x^{4} + 4 x^{2} + 4^{2}) = 0$

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

$(x + 2) (x - 2) (x^{4} + 4 x^{2} + 16) = 0$

Factor.

Rewrite $x^{4} + 4 x^{2} + 16$ in a factored form.

Rewrite the middle term.

$(x + 2) (x - 2) (x^{4} + 2 x^{2} \cdot 4 - 4 x^{2} + 16) = 0$

Rearrange terms.

$(x + 2) (x - 2) (x^{4} + 2 x^{2} \cdot 4 + 16 - 4 x^{2}) = 0$

Factor first three terms by perfect square rule.

$(x + 2) (x - 2) ({(x^{2} + 4)}^{2} - 4 x^{2}) = 0$

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$(x + 2) (x - 2) ({(x^{2} + 4)}^{2} - {(2 x)}^{2}) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2} + 4$ and $b = 2 x$ .

$(x + 2) (x - 2) ((x^{2} + 4 + 2 x) (x^{2} + 4 - (2 x))) = 0$

Simplify.

Reorder terms.

$(x + 2) (x - 2) ((x^{2} + 2 x + 4) (x^{2} + 4 - (2 x))) = 0$

Multiply $2$ by $- 1$ .

$(x + 2) (x - 2) ((x^{2} + 2 x + 4) (x^{2} + 4 - 2 x)) = 0$

Reorder terms.

$(x + 2) (x - 2) ((x^{2} + 2 x + 4) (x^{2} - 2 x + 4)) = 0$

Remove unnecessary parentheses.

$(x + 2) (x - 2) (x^{2} + 2 x + 4) (x^{2} - 2 x + 4) = 0$

Factor x^6-64=0

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