Simplify i^89

Problem

i^89

Solution

1. Identify the cycle of the imaginary unit i The powers of i repeat every four terms: i^1=i i^2=−1 i^3=−i and i^4=1

2. Divide the exponent by 4 to find the remainder. We calculate 89÷4

3. Calculate the quotient and remainder. Since 89=4×22+1 the remainder is 1

4. Apply the property i^n=i^r where r is the remainder of n÷4

5. Substitute the remainder into the expression to get i^1

Final Answer

i^89=i

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