Basic Math Examples

$Z^{6} + 1$

Step 1

Rewrite $Z^{6}$ as ${(Z^{2})}^{3}$ .

${(Z^{2})}^{3} + 1$

Step 2

Rewrite $1$ as $1^{3}$ .

${(Z^{2})}^{3} + 1^{3}$

Step 3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = Z^{2}$ and $b = 1$ .

$(Z^{2} + 1) ({(Z^{2})}^{2} - Z^{2} \cdot 1 + 1^{2})$

Step 4

Simplify.

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Step 4.1

Multiply the exponents in ${(Z^{2})}^{2}$ .

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Step 4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(Z^{2} + 1) (Z^{2 \cdot 2} - Z^{2} \cdot 1 + 1^{2})$

Step 4.1.2

Multiply $2$ by $2$ .

$(Z^{2} + 1) (Z^{4} - Z^{2} \cdot 1 + 1^{2})$

Step 4.2

Multiply $- 1$ by $1$ .

$(Z^{2} + 1) (Z^{4} - Z^{2} + 1^{2})$

Step 4.3

One to any power is one.

$(Z^{2} + 1) (Z^{4} - Z^{2} + 1)$