Simplify i^15

Problem

i15

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.

15÷4=3* remainder *3

3. Rewrite the expression using the property of exponents in=i(4*k+r)=(i4)k⋅ir

i15=i12⋅i3

4. Substitute the known values i12=(i4)3=1=1 and i3=−i

i15=1⋅(−i)

5. Simplify the final result.

i15=−i

Final Answer

i15=−i

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