Simplify i^85-9i^76-8i^85+5i^111

$i^{85} - 9 i^{76} - 8 i^{85} + 5 i^{111}$

Step 1

Simplify each term.

Rewrite $i^{85}$ as ${(i^{4})}^{21} i$ .

Factor out $i^{84}$ .

$i^{84} i - 9 i^{76} - 8 i^{85} + 5 i^{111}$

Rewrite $i^{84}$ as ${(i^{4})}^{21}$ .

${(i^{4})}^{21} i - 9 i^{76} - 8 i^{85} + 5 i^{111}$

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

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

${({(i^{2})}^{2})}^{21} i - 9 i^{76} - 8 i^{85} + 5 i^{111}$

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

${({(- 1)}^{2})}^{21} i - 9 i^{76} - 8 i^{85} + 5 i^{111}$

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

$1^{21} i - 9 i^{76} - 8 i^{85} + 5 i^{111}$

One to any power is one.

$1 i - 9 i^{76} - 8 i^{85} + 5 i^{111}$

Multiply $i$ by $1$ .

$i - 9 i^{76} - 8 i^{85} + 5 i^{111}$

Rewrite $i^{76}$ as ${(i^{4})}^{19}$ .

$i - 9 {(i^{4})}^{19} - 8 i^{85} + 5 i^{111}$

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

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

$i - 9 {({(i^{2})}^{2})}^{19} - 8 i^{85} + 5 i^{111}$

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

$i - 9 {({(- 1)}^{2})}^{19} - 8 i^{85} + 5 i^{111}$

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

$i - 9 \cdot 1^{19} - 8 i^{85} + 5 i^{111}$

One to any power is one.

$i - 9 \cdot 1 - 8 i^{85} + 5 i^{111}$

Multiply $- 9$ by $1$ .

$i - 9 - 8 i^{85} + 5 i^{111}$

Rewrite $i^{85}$ as ${(i^{4})}^{21} i$ .

Factor out $i^{84}$ .

$i - 9 - 8 (i^{84} i) + 5 i^{111}$

Rewrite $i^{84}$ as ${(i^{4})}^{21}$ .

$i - 9 - 8 ({(i^{4})}^{21} i) + 5 i^{111}$

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

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

$i - 9 - 8 ({({(i^{2})}^{2})}^{21} i) + 5 i^{111}$

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

$i - 9 - 8 ({({(- 1)}^{2})}^{21} i) + 5 i^{111}$

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

$i - 9 - 8 (1^{21} i) + 5 i^{111}$

One to any power is one.

$i - 9 - 8 (1 i) + 5 i^{111}$

Multiply $i$ by $1$ .

$i - 9 - 8 i + 5 i^{111}$

Rewrite $i^{111}$ as ${(i^{4})}^{27} (i^{2} \cdot i)$ .

Factor out $i^{108}$ .

$i - 9 - 8 i + 5 (i^{108} i^{3})$

Rewrite $i^{108}$ as ${(i^{4})}^{27}$ .

$i - 9 - 8 i + 5 ({(i^{4})}^{27} i^{3})$

Factor out $i^{2}$ .

$i - 9 - 8 i + 5 ({(i^{4})}^{27} (i^{2} \cdot i))$

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

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

$i - 9 - 8 i + 5 ({({(i^{2})}^{2})}^{27} (i^{2} \cdot i))$

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

$i - 9 - 8 i + 5 ({({(- 1)}^{2})}^{27} (i^{2} \cdot i))$

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

$i - 9 - 8 i + 5 (1^{27} (i^{2} \cdot i))$

One to any power is one.

$i - 9 - 8 i + 5 (1 (i^{2} \cdot i))$

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

$i - 9 - 8 i + 5 (i^{2} \cdot i)$

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

$i - 9 - 8 i + 5 (- 1 \cdot i)$

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

$i - 9 - 8 i + 5 (- i)$

Multiply $- 1$ by $5$ .

$i - 9 - 8 i - 5 i$

Step 2

Simplify by adding terms.

Subtract $8 i$ from $i$ .

$- 9 - 7 i - 5 i$

Subtract $5 i$ from $- 7 i$ .

$- 9 - 12 i$

Simplify i^85-9i^76-8i^85+5i^111

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