Simplify i^50

Problem

i50

Solution

1. Identify the cycle of powers of the imaginary unit i The powers of i repeat every four terms: i1=i i2=−1 i3=−i and i4=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 in=ir where r is the remainder of n÷4

i50=i2

5. Substitute the known value of i2

i2=−1

Final Answer

i50=−1

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