Simplify i^67

Problem

i^67

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 find the remainder.

67=4×16+3

3. Rewrite the expression using the properties of exponents.

i^67=i^(4×16+3)

i^67=(i^4)^16×i^3

4. Substitute the known values i^4=1 and i^3=−i into the expression.

i^67=(1)^16×(−i)

5. Simplify the final result.

i^67=1×(−i)

i^67=−i

Final Answer

i^67=−i

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