Simplify i^48

Problem

i^48

Solution

1. Identify the cycle 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 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 i^n where n is a multiple of 4 is equal to i^4 raised to some power.

i^48=(i^4)^12

5. Substitute the value i^4=1 into the expression.

1^12=1

Final Answer

i^48=1

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