Simplify i^43

Problem

i43

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 43÷4

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

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

5. Substitute the value of i3 into the expression.

Final Answer

i43=−i

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