Question

simplify the power of $i$.
$i^{40}=$

Explanation:

Step1: Recall the powers of $i$ cycle

The powers of $i$ have a cycle: $i^1 = i$, $i^2=- 1$, $i^3 = i^2\times i=-i$, $i^4=(i^2)^2 = 1$.

Step2: Divide the exponent by 4

Divide 40 by 4: $40\div4 = 10$ with a remainder of 0.

Step3: Determine the value

Since the remainder is 0 when 40 is divided by 4, $i^{40}=(i^4)^{10}$. And since $i^4 = 1$, then $(i^4)^{10}=1^{10}=1$.

Answer:

1