Precalculus Examples

$x^{9} - 64 y^{12}$

Step 1

Rewrite $x^{9}$ as ${(x^{3})}^{3}$ .

${(x^{3})}^{3} - 64 y^{12}$

Step 2

Rewrite $64 y^{12}$ as ${(4 y^{4})}^{3}$ .

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

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 $a = x^{3}$ and $b = 4 y^{4}$ .

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

Step 4

Simplify.

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Step 4.1

Multiply $4$ by $- 1$ .

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

Step 4.2

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 4.2.1

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

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

Step 4.2.2

Multiply $3$ by $2$ .

$(x^{3} - 4 y^{4}) (x^{6} + x^{3} (4 y^{4}) + {(4 y^{4})}^{2})$

Step 4.3

Rewrite using the commutative property of multiplication.

$(x^{3} - 4 y^{4}) (x^{6} + 4 x^{3} y^{4} + {(4 y^{4})}^{2})$

Step 4.4

Apply the product rule to $4 y^{4}$ .

$(x^{3} - 4 y^{4}) (x^{6} + 4 x^{3} y^{4} + 4^{2} {(y^{4})}^{2})$

Step 4.5

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

$(x^{3} - 4 y^{4}) (x^{6} + 4 x^{3} y^{4} + 16 {(y^{4})}^{2})$

Step 4.6

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

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Step 4.6.1

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

$(x^{3} - 4 y^{4}) (x^{6} + 4 x^{3} y^{4} + 16 y^{4 \cdot 2})$

Step 4.6.2

Multiply $4$ by $2$ .

$(x^{3} - 4 y^{4}) (x^{6} + 4 x^{3} y^{4} + 16 y^{8})$