Question

complex numbers
directions: simplify operations with complex numbers

1. $i^{34}$
2. $i^{146}$
3. $i^{68}$
4. $i^{635}$

helpful review:
$i^1 = \underline{qquadqquadqquadqquad} = \underline{qquadqquad}$
$i^2 = \underline{qquadqquadqquadqquad} = \underline{qquadqquad}$
$i^3 = \underline{qquadqquadqquadqquad} = \underline{qquadqquad}$
$i^4 = \underline{qquadqquadqquadqquad} = \underline{qquadqquad}$

Explanation:

Step1: Recall $i$ cycle rules

$i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$; cycle repeats every 4.

Step2: Simplify $i^{34}$

Divide exponent by 4: $34\div4=8$ remainder 2. So $i^{34}=i^2=-1$.

Step3: Simplify $i^{146}$

Divide exponent by 4: $146\div4=36$ remainder 2. So $i^{146}=i^2=-1$.

Step4: Simplify $i^{68}$

Divide exponent by 4: $68\div4=17$ remainder 0. So $i^{68}=i^4=1$.

Step5: Simplify $i^{635}$

Divide exponent by 4: $635\div4=158$ remainder 3. So $i^{635}=i^3=-i$.

Step6: Fill review values

$i^1 = i$; $i^2 = \sqrt{-1}\times\sqrt{-1} = -1$; $i^3 = i^2\times i = -i$; $i^4 = i^2\times i^2 = 1$.

Answer:

1. $\boldsymbol{-1}$
2. $\boldsymbol{-1}$
3. $\boldsymbol{1}$
4. $\boldsymbol{-i}$

Helpful review:
$i^1 = \boldsymbol{i} = \boldsymbol{i}$
$i^2 = \boldsymbol{\sqrt{-1} \times \sqrt{-1}} = \boldsymbol{-1}$
$i^3 = \boldsymbol{i^2 \times i} = \boldsymbol{-i}$
$i^4 = \boldsymbol{i^2 \times i^2} = \boldsymbol{1}$