What is the value of `-i^(48)`?

Options

A. `-i`
B. `i`
C. `-1`
D. `1`

Answer

To find the value of \(-i^{48}\), we can follow these steps:

### Step 1: Understand the properties of \(i\)
We know that:
- \(i = \sqrt{-1}\)
- \(i^2 = -1\)
- \(i^3 = i^2 \cdot i = -1 \cdot i = -i\)
- \(i^4 = (i^2)^2 = (-1)^2 = 1\)

### Step 2: Identify the pattern in powers of \(i\)
The powers of \(i\) repeat every four terms:
- \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\)
- \(i^5 = i\), and so on.

### Step 3: Simplify \(-i^{48}\)
Since \(i\) has a periodicity of 4, we can reduce \(48\) modulo \(4\):
\[ 48 \mod 4 = 0 \]
This means:
\[ i^{48} = (i^4)^{12} = 1^{12} = 1 \]

### Step 4: Substitute back into the expression
Now we can substitute this back into our expression for \(-i^{48}\):
\[ -i^{48} = -1 \]

### Conclusion
Thus, the value of \(-i^{48}\) is \(-1\).