Simplify i^55

Problem

i55

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. We calculate 55÷4

3. Calculate the quotient and remainder. Since 4×13=52 we have 55 = 4(13) + 3$.

4. Apply the property of exponents i(4*n+r)=ir In this case, i55=i(4*(13)+3)=(i4)13⋅i3

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

6. Simplify the result to find the final value.

Final Answer

i55=−i

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