Simplify i^16

Problem

i16

Solution

1. Identify the cycle 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.

16÷4=4

3. Determine the remainder of the division. Since 16 is exactly divisible by 4, the remainder is 0.

16=4×4+0

4. Apply the property that i(4*k)=(i4)k=1 for any integer k

i16=(i4)4

5. Substitute the value i4=1 into the expression.

1=1

Final Answer

i16=1

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