Simplify i^8+i^9+i^10+i^11

$i^{8} + i^{9} + i^{10} + i^{11}$

Simplify each term.

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

${(i^{4})}^{2} + i^{9} + i^{10} + i^{11}$

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

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

${({(i^{2})}^{2})}^{2} + i^{9} + i^{10} + i^{11}$

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

${({(- 1)}^{2})}^{2} + i^{9} + i^{10} + i^{11}$

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

$1^{2} + i^{9} + i^{10} + i^{11}$

$1^{2} + i^{9} + i^{10} + i^{11}$

One to any power is one.

$1 + i^{9} + i^{10} + i^{11}$

Rewrite $i^{9}$ as ${(i^{4})}^{2} i$ .

Factor out $i^{8}$ .

$1 + i^{8} i + i^{10} + i^{11}$

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

$1 + {(i^{4})}^{2} i + i^{10} + i^{11}$

$1 + {(i^{4})}^{2} i + i^{10} + i^{11}$

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

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

$1 + {({(i^{2})}^{2})}^{2} i + i^{10} + i^{11}$

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

$1 + {({(- 1)}^{2})}^{2} i + i^{10} + i^{11}$

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

$1 + 1^{2} i + i^{10} + i^{11}$

$1 + 1^{2} i + i^{10} + i^{11}$

One to any power is one.

$1 + 1 i + i^{10} + i^{11}$

Multiply $i$ by $1$ .

$1 + i + i^{10} + i^{11}$

Rewrite $i^{10}$ as ${(i^{4})}^{2} i^{2}$ .

Factor out $i^{8}$ .

$1 + i + i^{8} i^{2} + i^{11}$

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

$1 + i + {(i^{4})}^{2} i^{2} + i^{11}$

$1 + i + {(i^{4})}^{2} i^{2} + i^{11}$

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

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

$1 + i + {({(i^{2})}^{2})}^{2} i^{2} + i^{11}$

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

$1 + i + {({(- 1)}^{2})}^{2} i^{2} + i^{11}$

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

$1 + i + 1^{2} i^{2} + i^{11}$

$1 + i + 1^{2} i^{2} + i^{11}$

One to any power is one.

$1 + i + 1 i^{2} + i^{11}$

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

$1 + i + i^{2} + i^{11}$

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

$1 + i - 1 + i^{11}$

Rewrite $i^{11}$ as ${(i^{4})}^{2} (i^{2} \cdot i)$ .

Factor out $i^{8}$ .

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

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

$1 + i - 1 + {(i^{4})}^{2} i^{3}$

Factor out $i^{2}$ .

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

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

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

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

$1 + i - 1 - 1 \cdot i$

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

$1 + i - 1 - i$

$1 + i - 1 - i$

Simplify by adding terms.

Subtract $1$ from $1$ .

$0 + i - i$

Add $0$ and $i$ .

$i - i$

Subtract $i$ from $i$ .

$0$

$0$

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