Precalculus Examples

$64 x^{3} y^{6} - 27 z^{9} w^{3}$

Step 1

Rewrite $64 x^{3} y^{6}$ as ${(4 x y^{2})}^{3}$ .

${(4 x y^{2})}^{3} - 27 z^{9} w^{3}$

Step 2

Rewrite $27 z^{9} w^{3}$ as ${(3 z^{3} w)}^{3}$ .

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

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 $a = 4 x y^{2}$ and $b = 3 z^{3} w$ .

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

Step 4

Simplify.

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Step 4.1

Multiply $3$ by $- 1$ .

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

Step 4.2

Remove parentheses.

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

Step 4.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.1

Apply the product rule to $4 x y^{2}$ .

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

Step 4.3.2

Apply the product rule to $4 x$ .

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

Step 4.4

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Multiply $2$ by $2$ .

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

Step 4.6

Multiply $3$ by $4$ .

$(4 x y^{2} - 3 z^{3} w) (16 x^{2} y^{4} + 12 x y^{2} (z^{3} w) + {(3 z^{3} w)}^{2})$

Step 4.7

Remove parentheses.

$(4 x y^{2} - 3 z^{3} w) (16 x^{2} y^{4} + 12 x y^{2} z^{3} w + {(3 z^{3} w)}^{2})$

Step 4.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.8.1

Apply the product rule to $3 z^{3} w$ .

$(4 x y^{2} - 3 z^{3} w) (16 x^{2} y^{4} + 12 x y^{2} z^{3} w + {(3 z^{3})}^{2} w^{2})$

Step 4.8.2

Apply the product rule to $3 z^{3}$ .

$(4 x y^{2} - 3 z^{3} w) (16 x^{2} y^{4} + 12 x y^{2} z^{3} w + 3^{2} {(z^{3})}^{2} w^{2})$

Step 4.9

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

$(4 x y^{2} - 3 z^{3} w) (16 x^{2} y^{4} + 12 x y^{2} z^{3} w + 9 {(z^{3})}^{2} w^{2})$

Step 4.10

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

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Step 4.10.1

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

$(4 x y^{2} - 3 z^{3} w) (16 x^{2} y^{4} + 12 x y^{2} z^{3} w + 9 z^{3 \cdot 2} w^{2})$

Step 4.10.2

Multiply $3$ by $2$ .

$(4 x y^{2} - 3 z^{3} w) (16 x^{2} y^{4} + 12 x y^{2} z^{3} w + 9 z^{6} w^{2})$