Simplifying Powers of i

Evaluate: $i^{35} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Since $i^{4} = 1,$ we can simplify the problem by factoring out as many factors of $i^{4}$ as possible. To do so, first determine how many times 4 goes into 35: $35 = 4 \cdot 8 + 3.$

$i^{35} = i^{4 \cdot 8 + 3} = i^{4 \cdot 8} \cdot i^{3} = {(i^{4})}^{8} \cdot i^{3} = 1^{8} \cdot i^{3} = i^{3} = - i$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Evaluate: $i^{18}$

How did it go?