Simplify i^45

Problem

i^45

Solution

1. Identify the cycle of the powers of the imaginary unit i The values repeat every four powers: i^1=i i^2=−1 i^3=−i and i^4=1

2. Divide the exponent by 4 to find the remainder.

45÷4=11* remainder *1

3. Rewrite the expression using the property of exponents i^n=i^(4*k+r)=(i^4)^k⋅i^r

i^45=i^(4*(11)+1)

4. Substitute the known value i^4=1 into the expression.

i^45=(i^4)^11⋅i^1

5. Simplify the result using the fact that 1^11=1

i^45=1⋅i

i^45=i

Final Answer

i^45=i

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