Simplify i^5+i^6+i^7+i^8

$i^{5} + i^{6} + i^{7} + i^{8}$

Simplify each term.

Factor out $i^{4}$ .

$i^{4} i + i^{6} + i^{7} + i^{8}$

Rewrite $i^{4}$ as $1$ .

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

${(i^{2})}^{2} i + i^{6} + i^{7} + i^{8}$

Rewrite $i^{2}$ as $- 1$ .

${(- 1)}^{2} i + i^{6} + i^{7} + i^{8}$

Raise $- 1$ to the power of $2$ .

$1 i + i^{6} + i^{7} + i^{8}$

$1 i + i^{6} + i^{7} + i^{8}$

Multiply $i$ by $1$ .

$i + i^{6} + i^{7} + i^{8}$

Factor out $i^{4}$ .

$i + i^{4} i^{2} + i^{7} + i^{8}$

Rewrite $i^{4}$ as $1$ .

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$i + {(i^{2})}^{2} i^{2} + i^{7} + i^{8}$

Rewrite $i^{2}$ as $- 1$ .

$i + {(- 1)}^{2} i^{2} + i^{7} + i^{8}$

Raise $- 1$ to the power of $2$ .

$i + 1 i^{2} + i^{7} + i^{8}$

$i + 1 i^{2} + i^{7} + i^{8}$

Multiply $i^{2}$ by $1$ .

$i + i^{2} + i^{7} + i^{8}$

Rewrite $i^{2}$ as $- 1$ .

$i - 1 + i^{7} + i^{8}$

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

Factor out $i^{4}$ .

$i - 1 + i^{4} i^{3} + i^{8}$

Factor out $i^{2}$ .

$i - 1 + i^{4} (i^{2} \cdot i) + i^{8}$

$i - 1 + i^{4} (i^{2} \cdot i) + i^{8}$

Rewrite $i^{4}$ as $1$ .

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$i - 1 + {(i^{2})}^{2} (i^{2} \cdot i) + i^{8}$

Rewrite $i^{2}$ as $- 1$ .

$i - 1 + {(- 1)}^{2} (i^{2} \cdot i) + i^{8}$

Raise $- 1$ to the power of $2$ .

$i - 1 + 1 (i^{2} \cdot i) + i^{8}$

$i - 1 + 1 (i^{2} \cdot i) + i^{8}$

Multiply $i^{2} \cdot i$ by $1$ .

$i - 1 + i^{2} \cdot i + i^{8}$

Rewrite $i^{2}$ as $- 1$ .

$i - 1 - 1 \cdot i + i^{8}$

Rewrite $- 1 i$ as $- i$ .

$i - 1 - i + i^{8}$

Rewrite $i^{8}$ as ${(i^{4})}^{2}$ .

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

Rewrite $i^{4}$ as $1$ .

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$i - 1 - i + {({(i^{2})}^{2})}^{2}$

Rewrite $i^{2}$ as $- 1$ .

$i - 1 - i + {({(- 1)}^{2})}^{2}$

Raise $- 1$ to the power of $2$ .

$i - 1 - i + 1^{2}$

$i - 1 - i + 1^{2}$

One to any power is one.

$i - 1 - i + 1$

$i - 1 - i + 1$

Simplify by adding terms.

Subtract $i$ from $i$ .

$0 - 1 + 1$

Simplify by adding and subtracting.

Subtract $1$ from $0$ .

$- 1 + 1$

Add $- 1$ and $1$ .

$0$

$0$

$0$

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