Simplify i^31

Problem

i31

Solution

1. Identify the cycle of powers 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.

31÷4=7* remainder *3

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

i31=i(4*(7)+3)

4. Substitute the known values i4=1 and i3=−i into the expression.

i31=(i4)7⋅i3

i31=(1)7⋅(−i)

5. Simplify the final result.

i31=1⋅(−i)

i31=−i

Final Answer

i31=−i

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