simplify each of the following powers of $i$. $i^{15}=$

Explanation:

Step1: Recall the pattern of powers of i

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

Step2: Divide the exponent by 4

Divide 15 by 4: (15\div4 = 3) with a remainder of 3. That is (15 = 4\times3+3).

Step3: Rewrite the power of i

(i^{15}=i^{4\times3 + 3}=(i^4)^3\times i^3). Since (i^4 = 1), then ((i^4)^3=1^3 = 1). And (i^3=-i). So ((i^4)^3\times i^3=1\times(-i)=-i).

Answer:

(-i)