Basic Math Examples

a^{7} + a b^{6}

$a^{7} + a b^{6}$

Step 1

Factor

a

$a$ out of

a^{7} + a b^{6}

$a^{7} + a b^{6}$ .

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Step 1.1

Factor

a

$a$ out of

a^{7}

$a^{7}$ .

a \cdot a^{6} + a b^{6}

$a \cdot a^{6} + a b^{6}$

Step 1.2

Factor

a

$a$ out of

a b^{6}

$a b^{6}$ .

a \cdot a^{6} + a (b^{6})

$a \cdot a^{6} + a (b^{6})$

Step 1.3

Factor

a

$a$ out of

a \cdot a^{6} + a (b^{6})

$a \cdot a^{6} + a (b^{6})$ .

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

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

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

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

Step 2

Rewrite

a^{6}

$a^{6}$ as

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

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

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

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

Step 3

Rewrite

b^{6}

$b^{6}$ as

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

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

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

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

Step 4

Since both terms are perfect cubes, factor using the sum 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 ((a^{2} + b^{2}) ({(a^{2})}^{2} - a^{2} b^{2} + {(b^{2})}^{2}))

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Multiply the exponents in

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

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

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Step 5.1.1.1

Apply the power rule and multiply exponents,

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

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

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

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

Step 5.1.1.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 5.1.2

Multiply the exponents in

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

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

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Step 5.1.2.1

Apply the power rule and multiply exponents,

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

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

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

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

Step 5.1.2.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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

Step 5.2

Remove unnecessary parentheses.

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

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

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

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