Question

question
evaluate:
$i^{33}$
answer
$-i$ $1$
$-1$ $i$

Explanation:

Step1: Recall the property of imaginary unit \(i\)

The imaginary unit \(i\) has a cyclic property: \(i^1 = i\), \(i^2=- 1\), \(i^3=i^2\times i=-i\), \(i^4=(i^2)^2 = (-1)^2 = 1\), and then the cycle repeats every 4 powers. So we can express the exponent 33 as a multiple of 4 plus a remainder.

Step2: Divide 33 by 4

We divide 33 by 4: \(33\div4 = 8\) with a remainder of 1. Mathematically, \(33=4\times8 + 1\).

Step3: Rewrite \(i^{33}\) using the cyclic property

Using the property \(a^{m + n}=a^m\times a^n\) and \(a^{mn}=(a^m)^n\), we can rewrite \(i^{33}\) as \(i^{4\times8+1}=(i^4)^8\times i^1\).

Step4: Substitute the value of \(i^4\)

Since \(i^4 = 1\), then \((i^4)^8=1^8 = 1\). And \(i^1=i\). So \((i^4)^8\times i^1=1\times i=i\).

Answer:

\(i\) (corresponding to the option with \(i\))