The product of cube roots of –1 is equal :

Options

A. 0
B. 1
C. `-1`
D. None of these

Answer

To solve the problem of finding the product of the cube roots of -1, we can follow these steps:

### Step-by-Step Solution:

1. **Set Up the Equation**:
We want to find the cube roots of -1. We can express this mathematically as:
\[ z^3 = -1 \]

2. **Rewrite the Equation**:
We can rewrite this equation as:
\[ z^3 + 1 = 0 \]

3. **Identify Coefficients**:
In the polynomial \( z^3 + 1 = 0 \), we can identify the coefficients:
- \( a = 1 \) (coefficient of \( z^3 \))
- \( d = 1 \) (constant term)

4. **Use the Product of Roots Formula**:
The product of the roots of a polynomial \( ax^n + bx^{n-1} + ... + d = 0 \) is given by:
\[ \text{Product of roots} = (-1)^n \cdot \frac{d}{a} \]
Here, \( n = 3 \) (since it's a cubic equation), so:
\[ \text{Product of roots} = (-1)^3 \cdot \frac{1}{1} = -1 \]

5. **Conclusion**:
Therefore, the product of the cube roots of -1 is:
\[ z_1 \cdot z_2 \cdot z_3 = -1 \]

### Final Answer:
The product of the cube roots of -1 is equal to **-1**.

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