Question

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 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 15 by 4. \(15\div4 = 3\) with a remainder of 3. So \(i^{15}=i^{4\times3 + 3}\).

Step3: Use the exponent - rule \(a^{m + n}=a^m\times a^n\)

We know that \(i^{4\times3+3}=i^{4\times3}\times i^{3}\). Since \(i^{4k} = 1\) for any integer \(k\) (here \(k = 3\)), then \(i^{4\times3}=1\). And \(i^{3}=-i\). So \(i^{15}=-i\).

Answer:

\(-i\)