For the complex number i and an interger x, which of the following is a possible value of `i^(x)`?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem of finding a possible value of \( i^x \) where \( i \) is the complex number defined as \( i = \sqrt{-1} \) and \( x \) is an integer, we can follow these steps:

### Step 1: Understand the properties of \( i \)
The complex number \( i \) has specific powers that cycle through a set of values. We can calculate the first few powers of \( i \):
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = i^2 \cdot i = -1 \cdot i = -i \)
- \( i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1 \)

### Step 2: Identify the cycle of powers
From the calculations above, we can see that the powers of \( i \) repeat every four integers:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \) (and the cycle repeats)

### Step 3: Generalize the powers of \( i \)
We can express the powers of \( i \) based on the remainder when \( x \) is divided by 4:
- If \( x \equiv 0 \mod 4 \), then \( i^x = 1 \)
- If \( x \equiv 1 \mod 4 \), then \( i^x = i \)
- If \( x \equiv 2 \mod 4 \), then \( i^x = -1 \)
- If \( x \equiv 3 \mod 4 \), then \( i^x = -i \)

### Step 4: Determine possible values
From the above analysis, the possible values of \( i^x \) are:
- \( 1 \) (when \( x \) is a multiple of 4)
- \( i \) (when \( x \) gives a remainder of 1 when divided by 4)
- \( -1 \) (when \( x \) gives a remainder of 2 when divided by 4)
- \( -i \) (when \( x \) gives a remainder of 3 when divided by 4)

### Conclusion
Thus, the possible values of \( i^x \) for integer \( x \) are \( 1, i, -1, -i \).

If we are given options, we can select any of these values as a possible answer. For instance, if one of the options is \( 1 \), then that would be a correct choice.