Simplify i^89

Problem

i89

Solution

1. Identify the cycle of the imaginary unit i The powers of i repeat every four terms: i1=i i2=−1 i3=−i and i4=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 in=ir where r is the remainder of n÷4

5. Substitute the remainder into the expression to get i1

Final Answer

i89=i

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