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The question provided is asking to evaluate the truth of various statements based on the cube roots of \( a = 4 \) and \( b = -2 \).

We need to evaluate the expression:

\[
\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \frac{\sqrt[3]{4}}{\sqrt[3]{-2}}
\]

1. **Cube Root Properties**: 

   \[
   \frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}} = \sqrt[3]{\frac{4}{-2}} = \sqrt[3]{-2}
   \]

So, the expression simplifies to \( \sqrt[3]{-2} \).

Now let's evaluate the truth of the statements:

### Statement 1: \( \sqrt[3]{-2} \) is a rational number.
- **False**: Cube root of -2 is not a rational number (it cannot be expressed as the ratio of two integers).

### Statement 2: \( \sqrt[3]{-2} \) is not a natural number.
- **True**: Natural numbers are positive integers, and \( \sqrt[3]{-2} \) is neither an integer nor positive, so it is not a natural number.

### Statement 3: \( \sqrt[3]{-2} \) is an integer.
- **False**: As noted earlier, \( \sqrt[3]{-2} \) is not an integer.

### Statement 4: \( \sqrt[3]{-2} \) is an irrational number.
- **True**: Since \( \sqrt[3]{-2} \) cannot be written as a fraction, it is an irrational number.

### Conclusion:
- The correct answers are **Statement 2** and **Statement 4**.

Would you like further explanation on any of these points or help with similar problems?

Here are five related questions for practice:
1. What is the cube root of 8 divided by the cube root of -27?
2. Is \( \sqrt[3]{8/27} \) a rational or irrational number?
3. What is the cube root of -125? Is it rational or irrational?
4. How do cube roots of negative numbers differ from square roots of negative numbers?
5. What are the natural numbers, and why can't cube roots of negative numbers be natural?

**Tip**: Cube roots of negative numbers can result in negative values, unlike square roots which require imaginary numbers for negatives.

Let a = 4 and b = -2. Select ALL statements that are TRUE about the cube root of a divided by the cube root of b.

Explore how to evaluate the cube root of 4 divided by the cube root of -2. This problem covers cube roots and the classification of numbers into rational, irrational, natural numbers, and integers. Suitable for Grades 9-12.

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