Simplify i^43

Problem

i^43

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 43÷4

3. Determine the remainder. Since 43=(4×10)+3 the remainder is 3

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

5. Substitute the value of i^3 into the expression.

Final Answer

i^43=−i

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