Simplify i^24

Problem

i24

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 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)=(i4)k=1 for any integer k

i24=(i4)6

5. Evaluate the expression using the fact that i4=1

1=1

Final Answer

i24=1

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