Simplify i^24

Problem

i^24

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 determine the position in the cycle.

24÷4=6

3. Determine the remainder of the division. Since 24 is exactly divisible by 4, the remainder is 0.

24≡0*(mod()*4)

4. Apply the property that i^(4*k)=(i^4)^k=1^k for any integer k

i^24=(i^4)^6

5. Evaluate the expression using the fact that i^4=1

1^6=1

Final Answer

i^24=1

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