Simplify i^41

Problem

i41

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 41÷4

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

4. Apply the property in=ir where r is the remainder of n÷4

5. Simplify the expression using the remainder.

i41=i(4×10+1)

i41=(i4)10×i1

i41=1×i

i41=1×i

Final Answer

i41=i

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