Algebra Examples

$x^{6} - 64 y^{6}$

Step 1

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

${(x^{2})}^{3} - 64 y^{6}$

Step 2

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

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

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

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

Step 4

Simplify.

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Step 4.1

Remove parentheses.

$(x^{2} - 1 \cdot 4 y^{2}) ({(x^{2})}^{2} + x^{2} (4 y^{2}) + {(4 y^{2})}^{2})$

Step 4.2

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

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

Step 4.3

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 $b = 2 y$ .

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

Step 4.4

Multiply $2$ by $- 1$ .

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Multiply $2$ by $2$ .

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

Step 4.6

Rewrite using the commutative property of multiplication.

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

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

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Step 4.9.1

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

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

Step 4.9.2

Multiply $2$ by $2$ .

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

Step 4.10

Factor.

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Step 4.10.1

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

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Step 4.10.1.1

Rewrite the middle term.

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

Step 4.10.1.2

Rearrange terms.

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

Step 4.10.1.3

Factor first three terms by perfect square rule.

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

Step 4.10.1.4

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

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

Step 4.10.1.5

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

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

Step 4.10.1.6

Simplify.

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Step 4.10.1.6.1

Multiply $2$ by $- 1$ .

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

Step 4.10.1.6.2

Remove parentheses.

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

Step 4.10.2

Remove unnecessary parentheses.

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