Simplify i^20

Problem

i20

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 where the expression falls in the cycle.

20÷4=5

3. Determine the remainder of the division. Since 20 is exactly divisible by 4, the remainder is 0.

20=4×5+0

4. Apply the property i(4*k)=(i4)k=1

i20=(i4)5

5. Substitute i4=1 into the expression.

1=1

Final Answer

i20=1

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