Question

compute ( i^{512} ).

Explanation:

Step1: Record the cycle of \(i\)

We know that the imaginary unit \(i\) has a cycle of \(4\), that is:
\(i^1 = i\), \(i^2=- 1\), \(i^3=i^2\times i=-i\), \(i^4=(i^2)^2 = (-1)^2 = 1\), and then the powers of \(i\) will cycle every \(4\) times.

Step2: Calculate the remainder of the exponent divided by \(4\)

To calculate \(i^{512}\), we divide the exponent \(512\) by \(4\):
\(512\div4 = 128\), the remainder is \(0\).

Step3: Determine the value of \(i^{512}\)

When the remainder of the exponent divided by \(4\) is \(0\), it means \(i^{n}\) (where \(n = 512\)) is equal to \(i^{4k}\) (\(k = 128\) here), and according to the cycle property of \(i\), \(i^{4k}=(i^{4})^{k}\). Since \(i^{4} = 1\), then \((i^{4})^{k}=1^{k}=1\). So \(i^{512}=1\).

Answer:

\(1\)