Simplify i^45

Problem

i45

Solution

1. Identify the cycle of the powers of the imaginary unit i The values repeat every four powers: i1=i i2=−1 i3=−i and i4=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 in=i(4*k+r)=(i4)k⋅ir

i45=i(4*(11)+1)

4. Substitute the known value i4=1 into the expression.

i45=(i4)11⋅i1

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

i45=1⋅i

i45=i

Final Answer

i45=i

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