Algebra Examples

x^{6} - y^{6}

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

Step 1

Rewrite

x^{6}

$x^{6}$ as

{(x^{2})}^{3}

${(x^{2})}^{3}$ .

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

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

Step 2

Rewrite

y^{6}

$y^{6}$ as

{(y^{2})}^{3}

${(y^{2})}^{3}$ .

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

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

a = x^{2}

$a = x^{2}$ and

b = y^{2}

$b = y^{2}$ .

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

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = y

$b = y$ .

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

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

Step 4.2

Multiply the exponents in

{(x^{2})}^{2}

${(x^{2})}^{2}$ .

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Step 4.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

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

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

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

Step 4.2.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 4.3

Multiply the exponents in

{(y^{2})}^{2}

${(y^{2})}^{2}$ .

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Step 4.3.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

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

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

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

Step 4.3.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 4.4

Factor.

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Step 4.4.1

Rewrite

x^{4} + x^{2} y^{2} + y^{4}

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

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

Step 4.4.1.2

Rearrange terms.

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

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

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})

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

Step 4.4.1.4

Rewrite

x^{2} y^{2}

$x^{2} y^{2}$ as

{(x y)}^{2}

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

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

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x^{2} + y^{2}

$a = x^{2} + y^{2}$ and

b = x y

$b = x y$ .

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

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

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

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

Step 4.4.2

Remove unnecessary parentheses.

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

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

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

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

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

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