Evaluate v^81

Problem

i81

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 81÷4

3. Determine the remainder. Since 81=4×20+1 the remainder is 1

4. Simplify the expression using the property in=ir where r is the remainder of n÷4

5. Evaluate the resulting power: i1=i

Final Answer

i81=i

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