Simplify i^31

Problem

i^31

Solution

1. Identify the cycle of powers of the imaginary unit i The powers of i repeat every four terms: i^1=i i^2=−1 i^3=−i and i^4=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 i^n=i^(4*k+r)=(i^4)^k⋅i^r

i^31=i^(4*(7)+3)

4. Substitute the known values i^4=1 and i^3=−i into the expression.

i^31=(i^4)^7⋅i^3

i^31=(1)^7⋅(−i)

5. Simplify the final result.

i^31=1⋅(−i)

i^31=−i

Final Answer

i^31=−i

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