Precalculus Examples

$x^{6} + 729$

Step 1

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

${(x^{2})}^{3} + 729$

Step 2

Rewrite $729$ as $9^{3}$ .

${(x^{2})}^{3} + 9^{3}$

Step 3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x^{2}$ and $b = 9$ .

$(x^{2} + 9) ({(x^{2})}^{2} - x^{2} \cdot 9 + 9^{2})$

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

$(x^{2} + 9) (x^{2 \cdot 2} - x^{2} \cdot 9 + 9^{2})$

Step 4.1.2

Multiply $2$ by $2$ .

$(x^{2} + 9) (x^{4} - x^{2} \cdot 9 + 9^{2})$

Step 4.2

Multiply $9$ by $- 1$ .

$(x^{2} + 9) (x^{4} - 9 x^{2} + 9^{2})$

Step 4.3

Raise $9$ to the power of $2$ .

$(x^{2} + 9) (x^{4} - 9 x^{2} + 81)$