Precalculus Examples

$27 x^{3} - y^{6}$

Step 1

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

${(3 x)}^{3} - y^{6}$

Step 2

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

${(3 x)}^{3} - {(y^{2})}^{3}$

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 $a = 3 x$ and $b = y^{2}$ .

$(3 x - y^{2}) ({(3 x)}^{2} + 3 x y^{2} + {(y^{2})}^{2})$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $3 x$ .

$(3 x - y^{2}) (3^{2} x^{2} + 3 x y^{2} + {(y^{2})}^{2})$

Step 4.2

Raise $3$ to the power of $2$ .

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

Step 4.3

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

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Step 4.3.1

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

$(3 x - y^{2}) (9 x^{2} + 3 x y^{2} + y^{2 \cdot 2})$

Step 4.3.2

Multiply $2$ by $2$ .

$(3 x - y^{2}) (9 x^{2} + 3 x y^{2} + y^{4})$