Simplify i^22

Problem

i22

Solution

1. Identify the cycle of powers of the imaginary unit i

i1=i

i2=−1

i3=−i

i4=1

2. Divide the exponent by 4 to find the remainder, since the powers of i repeat every 4 terms.

22÷4=5* remainder *2

3. Rewrite the expression using the property of exponents and the fact that i4=1

i22=i(4*(5)+2)

i22=(i4)5⋅i2

4. Substitute the known values for i4 and i2

i22=(1)5⋅(−1)

5. Simplify the final result.

i22=1⋅(−1)

i22=−1

Final Answer

i22=−1

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