Algebra Examples

a^{6} + b^{6}

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

Step 1

Rewrite

a^{6}

$a^{6}$ as

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

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

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

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

Step 2

Rewrite

b^{6}

$b^{6}$ as

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

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

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply the exponents in

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

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

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Step 4.1.1

Apply the power rule and multiply exponents,

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

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

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

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

Step 4.1.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 4.2

Multiply the exponents in

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

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

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Step 4.2.1

Apply the power rule and multiply exponents,

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

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

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

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

Step 4.2.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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