Simplify i^27+i^34

$i^{27} + i^{34}$

Step 1

Simplify each term.

Rewrite $i^{27}$ as ${(i^{4})}^{6} (i^{2} \cdot i)$ .

Factor out $i^{24}$ .

$i^{24} i^{3} + i^{34}$

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

${(i^{4})}^{6} i^{3} + i^{34}$

Factor out $i^{2}$ .

${(i^{4})}^{6} (i^{2} \cdot i) + i^{34}$

${(i^{4})}^{6} (i^{2} \cdot i) + i^{34}$

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

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

${({(i^{2})}^{2})}^{6} (i^{2} \cdot i) + i^{34}$

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

${({(- 1)}^{2})}^{6} (i^{2} \cdot i) + i^{34}$

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

$1^{6} (i^{2} \cdot i) + i^{34}$

$1^{6} (i^{2} \cdot i) + i^{34}$

One to any power is one.

$1 (i^{2} \cdot i) + i^{34}$

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

$i^{2} \cdot i + i^{34}$

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

$- 1 \cdot i + i^{34}$

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

$- i + i^{34}$

Rewrite $i^{34}$ as ${(i^{4})}^{8} i^{2}$ .

Factor out $i^{32}$ .

$- i + i^{32} i^{2}$

Rewrite $i^{32}$ as ${(i^{4})}^{8}$ .

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

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

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

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

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

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

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

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

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

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

One to any power is one.

$- i + 1 i^{2}$

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

$- i + i^{2}$

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

$- i - 1$

$- i - 1$

Step 2

Reorder $- i$ and $- 1$ .

$- 1 - i$

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