Simplify i^41

Problem

i^41

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. We calculate 41÷4

3. Determine the quotient and remainder. Since 41=(4×10)+1 the remainder is 1

4. Apply the property i^n=i^r where r is the remainder of n÷4

5. Simplify the expression using the remainder.

i^41=i^(4×10+1)

i^41=(i^4)^10×i^1

i^41=1^10×i

i^41=1×i

Final Answer

i^41=i

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