Simplify i^48

Problem

i48

Solution

1. Identify the cycle 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 determine the remainder.

48÷4=12

3. Determine the remainder. Since 48 is exactly divisible by 4, the remainder is 0.

48=4×12+0

4. Apply the power rule for i Any power in where n is a multiple of 4 is equal to i4 raised to some power.

i48=(i4)12

5. Substitute the value i4=1 into the expression.

1=1

Final Answer

i48=1

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