Simplify i^50

Problem

i^50

Solution

1. Identify the cycle of 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. We calculate 50÷4

3. Determine the remainder. Since 50=4×12+2 the remainder is 2

4. Apply the property that i^n=i^r where r is the remainder of n÷4

i^50=i^2

5. Substitute the known value of i^2

i^2=−1

Final Answer

i^50=−1

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