Question

simplify the following expression.
$i^{-41}$
answer 2 points

Explanation:

Step1: Rewrite negative exponent

$i^{-41} = \frac{1}{i^{41}}$

Step2: Reduce exponent modulo 4

$41 = 4\times10 + 1$, so $i^{41}=i^{4\times10 + 1}=(i^4)^{10}\times i^1$

Step3: Use $i^4=1$ to simplify

$(i^4)^{10}\times i = 1^{10}\times i = i$

Step4: Rationalize the denominator

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

Step5: Use $i^2=-1$ to simplify

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

Answer:

$-i$