Simplify i^40

Problem

i^40

Solution

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

2. Recall that i^1=i i^2=−1 i^3=−i and i^4=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 i^n=i^r where r is the remainder of n÷4 Since the remainder is 0, we use the equivalent power i^4

i^40=(i^4)^10

6. Substitute the value i^4=1 into the expression.

1^10=1

Final Answer

i^40=1

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