Simplify i^60

Problem

i^60

Solution

1. Identify the cycle of the powers of the imaginary unit i The powers of i repeat every four terms: i^1=i i^2=−1 i^3=−i and i^4=1

2. Divide the exponent by 4 to determine the position in the cycle. We calculate 60÷4

3. Determine the remainder of the division. Since 60=4×15+0 the remainder is 0

4. Apply the rule for powers of i Any power i^n where n is a multiple of 4 (remainder 0) is equivalent to i^4

5. Evaluate the expression. Since the remainder is 0 i^60=(i^4)^15=1^15=1

Final Answer

i^60=1

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