Simplify i^7

Problem

i^7

Solution

1. Identify the cycle of powers of the imaginary unit i where i^1=i i^2=−1 i^3=−i and i^4=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 i^n=i^r where r is the remainder.

i^7=i^4⋅i^3

5. Substitute the known values i^4=1 and i^3=−i into the expression.

i^7=1⋅(−i)

Final Answer

i^7=−i

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