Simplify i^111-2i^49-i^118-8i^26

$i^{111} - 2 i^{49} - i^{118} - 8 i^{26}$

Step 1

Simplify each term.

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

Factor out $i^{108}$ .

$i^{108} i^{3} - 2 i^{49} - i^{118} - 8 i^{26}$

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

${(i^{4})}^{27} i^{3} - 2 i^{49} - i^{118} - 8 i^{26}$

Factor out $i^{2}$ .

${(i^{4})}^{27} (i^{2} \cdot i) - 2 i^{49} - i^{118} - 8 i^{26}$

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

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

${({(i^{2})}^{2})}^{27} (i^{2} \cdot i) - 2 i^{49} - i^{118} - 8 i^{26}$

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

${({(- 1)}^{2})}^{27} (i^{2} \cdot i) - 2 i^{49} - i^{118} - 8 i^{26}$

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

$1^{27} (i^{2} \cdot i) - 2 i^{49} - i^{118} - 8 i^{26}$

One to any power is one.

$1 (i^{2} \cdot i) - 2 i^{49} - i^{118} - 8 i^{26}$

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

$i^{2} \cdot i - 2 i^{49} - i^{118} - 8 i^{26}$

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

$- 1 \cdot i - 2 i^{49} - i^{118} - 8 i^{26}$

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

$- i - 2 i^{49} - i^{118} - 8 i^{26}$

Rewrite $i^{49}$ as ${(i^{4})}^{12} i$ .

Factor out $i^{48}$ .

$- i - 2 (i^{48} i) - i^{118} - 8 i^{26}$

Rewrite $i^{48}$ as ${(i^{4})}^{12}$ .

$- i - 2 ({(i^{4})}^{12} i) - i^{118} - 8 i^{26}$

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

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

$- i - 2 ({({(i^{2})}^{2})}^{12} i) - i^{118} - 8 i^{26}$

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

$- i - 2 ({({(- 1)}^{2})}^{12} i) - i^{118} - 8 i^{26}$

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

$- i - 2 (1^{12} i) - i^{118} - 8 i^{26}$

One to any power is one.

$- i - 2 (1 i) - i^{118} - 8 i^{26}$

Multiply $i$ by $1$ .

$- i - 2 i - i^{118} - 8 i^{26}$

Rewrite $i^{118}$ as ${(i^{4})}^{29} i^{2}$ .

Factor out $i^{116}$ .

$- i - 2 i - (i^{116} i^{2}) - 8 i^{26}$

Rewrite $i^{116}$ as ${(i^{4})}^{29}$ .

$- i - 2 i - ({(i^{4})}^{29} i^{2}) - 8 i^{26}$

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

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

$- i - 2 i - ({({(i^{2})}^{2})}^{29} i^{2}) - 8 i^{26}$

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

$- i - 2 i - ({({(- 1)}^{2})}^{29} i^{2}) - 8 i^{26}$

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

$- i - 2 i - (1^{29} i^{2}) - 8 i^{26}$

One to any power is one.

$- i - 2 i - (1 i^{2}) - 8 i^{26}$

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

$- i - 2 i - i^{2} - 8 i^{26}$

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

$- i - 2 i - - 1 - 8 i^{26}$

Multiply $- 1$ by $- 1$ .

$- i - 2 i + 1 - 8 i^{26}$

Rewrite $i^{26}$ as ${(i^{4})}^{6} i^{2}$ .

Factor out $i^{24}$ .

$- i - 2 i + 1 - 8 (i^{24} i^{2})$

Rewrite $i^{24}$ as ${(i^{4})}^{6}$ .

$- i - 2 i + 1 - 8 ({(i^{4})}^{6} i^{2})$

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

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

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

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

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

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

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

One to any power is one.

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

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

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

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

$- i - 2 i + 1 - 8 \cdot - 1$

Multiply $- 8$ by $- 1$ .

$- i - 2 i + 1 + 8$

Step 2

Simplify by adding terms.

Subtract $2 i$ from $- i$ .

$- 3 i + 1 + 8$

Simplify the expression.

Add $1$ and $8$ .

$- 3 i + 9$

Reorder $- 3 i$ and $9$ .

$9 - 3 i$

Simplify i^111-2i^49-i^118-8i^26

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