Understand i as a number with i^2=-1 and represent complex numbers as a+bi.
Simplify powers of $i$ using the repeating cycle $i, -1, -i, 1$
Problem
Evaluate \(i^{5}\) by writing \(5=4+1\) and using \(i^{4}=1\). State the resulting value from the \(\text{four}\text{-}\text{power}~\text{cycle}\).
Big Picture
What this problem is really about
Powers of the imaginary unit repeat in a four-step cycle, so the exponent’s remainder locates the value. We’ll build the first four powers, separate complete groups of four from the exponent, reduce the leftover factor, and check that the modulo-four position matches the cycle.
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