Simplify i^8-i^9+i^10

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

Step 1

Simplify each term.

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

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

Factor out $i^{8}$ .

$1 - (i^{8} i) + i^{10}$

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

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

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

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

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

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

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

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

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

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

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

One to any power is one.

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

Multiply $i$ by $1$ .

$1 - i + i^{10}$

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

Factor out $i^{8}$ .

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

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

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

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

$1 - i + i^{2}$

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

$1 - i - 1$

$1 - i - 1$

Step 2

Simplify by subtracting numbers.

Subtract $1$ from $1$ .

$0 - i$

Subtract $i$ from $0$ .

$- i$

$- i$

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