Question

compute ( i^{-217} ).

Explanation:

Step1: Recall $i^4=1$ rule

$i^4 = 1$, so $i^n$ cycles every 4.

Step2: Rewrite negative exponent

$i^{-217} = \frac{1}{i^{217}}$

Step3: Find exponent modulo 4

$217 \div 4 = 54$ remainder $1$, so $217 = 4\times54 +1$, thus $i^{217}=i^{4\times54+1}=(i^4)^{54}\times i^1=1^{54}\times i = i$

Step4: Simplify the fraction

$\frac{1}{i} = \frac{1}{i} \times \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i$

Answer:

$-i$