Evaluate i^13i^14i^22

$i^{13} \cdot i^{14} \cdot i^{22}$

Step 1

Rewrite $i^{13}$ as ${(i^{4})}^{3} i$ .

Factor out $i^{12}$ .

$i^{12} i \cdot i^{14} \cdot i^{22}$

Rewrite $i^{12}$ as ${(i^{4})}^{3}$ .

${(i^{4})}^{3} i \cdot i^{14} \cdot i^{22}$

${(i^{4})}^{3} i \cdot i^{14} \cdot i^{22}$

Step 2

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

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

${({(i^{2})}^{2})}^{3} i \cdot i^{14} \cdot i^{22}$

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

${({(- 1)}^{2})}^{3} i \cdot i^{14} \cdot i^{22}$

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

$1^{3} i \cdot i^{14} \cdot i^{22}$

$1^{3} i \cdot i^{14} \cdot i^{22}$

Step 3

One to any power is one.

$1 i \cdot i^{14} \cdot i^{22}$

Step 4

Multiply $i$ by $1$ .

$i \cdot i^{14} \cdot i^{22}$

Step 5

Rewrite $i^{14}$ as ${(i^{4})}^{3} i^{2}$ .

Factor out $i^{12}$ .

$i \cdot (i^{12} i^{2}) \cdot i^{22}$

Rewrite $i^{12}$ as ${(i^{4})}^{3}$ .

$i \cdot ({(i^{4})}^{3} i^{2}) \cdot i^{22}$

$i \cdot ({(i^{4})}^{3} i^{2}) \cdot i^{22}$

Step 6

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

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

$i \cdot ({({(i^{2})}^{2})}^{3} i^{2}) \cdot i^{22}$

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

$i \cdot ({({(- 1)}^{2})}^{3} i^{2}) \cdot i^{22}$

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

$i \cdot (1^{3} i^{2}) \cdot i^{22}$

$i \cdot (1^{3} i^{2}) \cdot i^{22}$

Step 7

One to any power is one.

$i \cdot (1 i^{2}) \cdot i^{22}$

Step 8

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

$i \cdot i^{2} \cdot i^{22}$

Step 9

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

$i \cdot - 1 \cdot i^{22}$

Step 10

Move $- 1$ to the left of $i$ .

$- 1 \cdot i \cdot i^{22}$

Step 11

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

$- i \cdot i^{22}$

Step 12

Rewrite $i^{22}$ as ${(i^{4})}^{5} i^{2}$ .

Factor out $i^{20}$ .

$- i \cdot (i^{20} i^{2})$

Rewrite $i^{20}$ as ${(i^{4})}^{5}$ .

$- i \cdot ({(i^{4})}^{5} i^{2})$

$- i \cdot ({(i^{4})}^{5} i^{2})$

Step 13

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

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

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

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

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

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

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

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

Step 14

One to any power is one.

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

Step 15

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

$- i \cdot i^{2}$

Step 16

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

$- i \cdot - 1$

Step 17

Multiply $- 1$ by $- 1$ .

$1 i$

Step 18

Multiply $i$ by $1$ .

$i$

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