Simplify i^9

Problem

i^9

Solution

1. Identify the cycle of the powers of the imaginary unit i

2. Recall that i^1=i i^2=−1 i^3=−i and i^4=1

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

9=4×2+1

4. Rewrite the expression using the properties of exponents.

i^9=i^(4×2+1)

i^9=(i^4)^2×i^1

5. Substitute the value i^4=1 into the expression.

i^9=(1)^2×i

6. Simplify the resulting value.

i^9=1×i

i^9=i

Final Answer

i^9=i

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