Simplify i^40

Problem

i40

Solution

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

2. Recall that i1=i i2=−1 i3=−i and i4=1

3. Divide the exponent by 4 to determine where it falls in the cycle.

40÷4=10

4. Determine the remainder of the division.

40*(mod()*4)=0

5. Apply the property that in=ir where r is the remainder of n÷4 Since the remainder is 0, we use the equivalent power i4

i40=(i4)10

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

1=1

Final Answer

i40=1

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