Which of the following in incorrect?

Options

A. The cube of an even natural number is always even.
B. The root of a rational number `x/y` is `root3(x)/root3(y)`.
C. The cube root of a negative number is always pisitive.
D. 2197 is a perfect cube.

Answer

To determine which of the following statements is incorrect, we will evaluate each option one by one.

### Step-by-Step Solution:

1. **Evaluate Option A: "Cube of an even number is always even."**
- Let's take an even number, for example, 8.
- Calculate the cube:
\[ 8^3 = 8 \times 8 \times 8 = 512 \]
- Since 512 is even, this statement is correct.
- **Conclusion:** This option is not incorrect.

2. **Evaluate Option B: "The cube root of a rational number \( \frac{x}{y} \) is \( \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \)."**
- The cube root of \( \frac{x}{y} \) can be expressed as:
\[ \sqrt[3]{\frac{x}{y}} = \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \]
- This statement is true for cube roots, hence this option is correct.
- **Conclusion:** This option is not incorrect.

3. **Evaluate Option C: "The cube root of a negative number is always positive."**
- Let's take a negative number, for example, -3.
- Calculate the cube:
\[ (-3)^3 = -3 \times -3 \times -3 = -27 \]
- The cube root of -27 is:
\[ \sqrt[3]{-27} = -3 \]
- Since the cube root of a negative number is negative, this statement is incorrect.
- **Conclusion:** This option is indeed incorrect.

4. **Evaluate Option D: "2197 is a perfect cube."**
- We need to check if 2197 is a perfect cube.
- Calculate the cube root:
\[ \sqrt[3]{2197} = 13 \quad (\text{since } 13^3 = 2197) \]
- Since 2197 is the cube of 13, this statement is correct.
- **Conclusion:** This option is not incorrect.

### Final Answer:
The incorrect statements are **C**: "The cube root of a negative number is always positive."