Which of the following is equal to `root(3)(-1)?`

Options

A. `(sqrt(3)+sqrt(-1))/( 2)`
B. `(-sqrt(3)+sqrt(-1))/(sqrt(-4))`
C. `(sqrt(3)-sqrt(-1))/(sqrt(-4))`
D. `-sqrt(-1)`

Explanation

To solve the problem of finding the cube root of \(-1\), we can follow these steps:

### Step-by-Step Solution:

1. **Set up the equation**:
Let \( x = \sqrt[3]{-1} \).
This means \( x^3 = -1 \).

2. **Rewrite the equation**:
We can rewrite this as:
\( x^3 + 1 = 0 \).

3. **Factor the equation**:
The equation \( x^3 + 1 \) can be factored using the sum of cubes formula:
\( x^3 + 1 = (x + 1)(x^2 - x + 1) = 0 \).

4. **Find the roots**:
Setting each factor to zero gives us:
- From \( x + 1 = 0 \):
\( x = -1 \)
- From \( x^2 - x + 1 = 0 \):
We can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -1, c = 1 \):
\( x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \)
\( = \frac{1 \pm \sqrt{1 - 4}}{2} \)
\( = \frac{1 \pm \sqrt{-3}}{2} \)
\( = \frac{1 \pm i\sqrt{3}}{2} \).

5. **Summarize the roots**:
The three cube roots of \(-1\) are:
\( x = -1 \),
\( x = \frac{1 + i\sqrt{3}}{2} \),
\( x = \frac{1 - i\sqrt{3}}{2} \).

### Conclusion:
Thus, the cube roots of \(-1\) are:
- \( -1 \)
- \( \frac{1 + i\sqrt{3}}{2} \)
- \( \frac{1 - i\sqrt{3}}{2} \)