Simplify i^7

Problem

i7

Solution

1. Identify the cycle of powers of the imaginary unit i where i1=i i2=−1 i3=−i and i4=1

2. Divide the exponent by 4 to find the remainder, since the powers of i repeat every 4 terms.

3. Calculate the remainder of 7÷4 which is 3

4. Rewrite the expression using the property in=ir where r is the remainder.

i7=i4⋅i3

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

i7=1⋅(−i)

Final Answer

i7=−i

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