Algebra Examples

a^{8} - a^{2} b^{6}

$a^{8} - a^{2} b^{6}$

Step 1

Factor out the GCF of

a^{2}

$a^{2}$ from

a^{8} - a^{2} b^{6}

$a^{8} - a^{2} b^{6}$ .

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Step 1.1

Factor out the GCF of

a^{2}

$a^{2}$ from each term in the polynomial.

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Step 1.1.1

Factor out the GCF of

a^{2}

$a^{2}$ from the expression

a^{8}

$a^{8}$ .

a^{2} (a^{6}) - a^{2} b^{6}

$a^{2} (a^{6}) - a^{2} b^{6}$

Step 1.1.2

Factor out the GCF of

a^{2}

$a^{2}$ from the expression

- a^{2} b^{6}

$- a^{2} b^{6}$ .

a^{2} (a^{6}) + a^{2} (- b^{6})

$a^{2} (a^{6}) + a^{2} (- b^{6})$

a^{2} (a^{6}) + a^{2} (- b^{6})

$a^{2} (a^{6}) + a^{2} (- b^{6})$

Step 1.2

Since all the terms share a common factor of

a^{2}

$a^{2}$ , it can be factored out of each term.

a^{2} (a^{6} - b^{6})

$a^{2} (a^{6} - b^{6})$

a^{2} (a^{6} - b^{6})

$a^{2} (a^{6} - b^{6})$

Step 2

Rewrite

a^{6}

$a^{6}$ as

{(a^{2})}^{3}

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

a^{2} ({(a^{2})}^{3} - b^{6})

$a^{2} ({(a^{2})}^{3} - b^{6})$

Step 3

Rewrite

b^{6}

$b^{6}$ as

{(b^{2})}^{3}

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

a^{2} ({(a^{2})}^{3} - {(b^{2})}^{3})

$a^{2} ({(a^{2})}^{3} - {(b^{2})}^{3})$

Step 4

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 = a^{2}

$a = a^{2}$ and

b = b^{2}

$b = b^{2}$ .

a^{2} ((a^{2} - b^{2}) ({(a^{2})}^{2} + a^{2} b^{2} + {(b^{2})}^{2}))

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

Step 5

Simplify.

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Step 5.1

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 = a

$a = a$ and

b = b

$b = b$ .

a^{2} ((a + b) (a - b) ({(a^{2})}^{2} + a^{2} b^{2} + {(b^{2})}^{2}))

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

Step 5.2

Multiply the exponents in

{(a^{2})}^{2}

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

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Step 5.2.1

Apply the power rule and multiply exponents,

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

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

a^{2} ((a + b) (a - b) (a^{2 \cdot 2} + a^{2} b^{2} + {(b^{2})}^{2}))

$a^{2} ((a + b) (a - b) (a^{2 \cdot 2} + a^{2} b^{2} + {(b^{2})}^{2}))$

Step 5.2.2

Multiply

2

$2$ by

2

$2$ .

a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + {(b^{2})}^{2}))

$a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + {(b^{2})}^{2}))$

a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + {(b^{2})}^{2}))

$a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + {(b^{2})}^{2}))$

Step 5.3

Multiply the exponents in

{(b^{2})}^{2}

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

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Step 5.3.1

Apply the power rule and multiply exponents,

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

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

a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + b^{2 \cdot 2}))

$a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + b^{2 \cdot 2}))$

Step 5.3.2

Multiply

2

$2$ by

2

$2$ .

a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + b^{4}))

$a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + b^{4}))$

a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + b^{4}))

$a^{2} ((a + b) (a - b) (a^{4} + a^{2} b^{2} + b^{4}))$

Step 5.4

Factor.

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Step 5.4.1

Rewrite

a^{4} + a^{2} b^{2} + b^{4}

$a^{4} + a^{2} b^{2} + b^{4}$ in a factored form.

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Step 5.4.1.1

Rewrite the middle term.

a^{2} ((a + b) (a - b) (a^{4} + 2 a^{2} b^{2} - a^{2} b^{2} + b^{4}))

$a^{2} ((a + b) (a - b) (a^{4} + 2 a^{2} b^{2} - a^{2} b^{2} + b^{4}))$

Step 5.4.1.2

Rearrange terms.

a^{2} ((a + b) (a - b) (a^{4} + 2 a^{2} b^{2} + b^{4} - a^{2} b^{2}))

$a^{2} ((a + b) (a - b) (a^{4} + 2 a^{2} b^{2} + b^{4} - a^{2} b^{2}))$

Step 5.4.1.3

Factor first three terms by perfect square rule.

a^{2} ((a + b) (a - b) ({(a^{2} + b^{2})}^{2} - a^{2} b^{2}))

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

Step 5.4.1.4

Rewrite

a^{2} b^{2}

$a^{2} b^{2}$ as

{(a b)}^{2}

${(a b)}^{2}$ .

a^{2} ((a + b) (a - b) ({(a^{2} + b^{2})}^{2} - {(a b)}^{2}))

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

Step 5.4.1.5

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 = a^{2} + b^{2}

$a = a^{2} + b^{2}$ and

b = a b

$b = a b$ .

a^{2} ((a + b) (a - b) ((a^{2} + b^{2} + a b) (a^{2} + b^{2} - (a b))))

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

Step 5.4.1.6

Remove parentheses.

a^{2} ((a + b) (a - b) ((a^{2} + b^{2} + a b) (a^{2} + b^{2} - a b)))

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

a^{2} ((a + b) (a - b) ((a^{2} + b^{2} + a b) (a^{2} + b^{2} - a b)))

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

Step 5.4.2

Remove unnecessary parentheses.

a^{2} ((a + b) (a - b) (a^{2} + b^{2} + a b) (a^{2} + b^{2} - a b))

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

a^{2} ((a + b) (a - b) (a^{2} + b^{2} + a b) (a^{2} + b^{2} - a b))

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

a^{2} ((a + b) (a - b) (a^{2} + b^{2} + a b) (a^{2} + b^{2} - a b))

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