Basic Math Examples

a^{6} - 8

$a^{6} - 8$

Step 1

Rewrite

a^{6}

$a^{6}$ as

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

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

{(a^{2})}^{3} - 8

${(a^{2})}^{3} - 8$

Step 2

Rewrite

8

$8$ as

2^{3}

$2^{3}$ .

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

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

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 = a^{2}

$a = a^{2}$ and

b = 2

$b = 2$ .

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

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

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

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

Step 4.1.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 4.2

Move

2

$2$ to the left of

a^{2}

$a^{2}$ .

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

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

Step 4.3

Raise

2

$2$ to the power of

2

$2$ .

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

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

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

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