Algebra Examples

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

Step 1

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

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

Step 2

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

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

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

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

Step 4

Simplify.

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Step 4.1

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 $b = y$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $2$ .

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.3.2

Multiply $2$ by $2$ .

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

Step 4.4

Factor.

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Step 4.4.1

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

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Step 4.4.1.1

Rewrite the middle term.

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

Step 4.4.1.2

Rearrange terms.

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

Step 4.4.1.3

Factor first three terms by perfect square rule.

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

Step 4.4.1.4

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

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

Step 4.4.1.5

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

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

Step 4.4.1.6

Remove parentheses.

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

Step 4.4.2

Remove unnecessary parentheses.

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