Simplify i^23

Problem

i^23

Solution

1. Identify the cycle 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.

23÷4=5* remainder *3

3. Rewrite the expression using the properties of exponents.

i^23=i^(4*(5)+3)

4. Apply the rule i^(4*n+r)=i^r

i^23=(i^4)^5⋅i^3

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

i^23=(1)^5⋅(−i)

6. Simplify the final result.

i^23=−i

Final Answer

i^23=−i

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