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The question is asking which of the provided expressions represent rational numbers. Let's analyze each expression:

1. \(\left(\sqrt{4}\right)^{\left(\sqrt{4}\right)}\)
2. \(\left(\sqrt[3]{2}\right)^{\left(\sqrt[3]{2}\right)}\)
3. \(\left(\sqrt{4}\right)^{\left(\sqrt[2]{4}\right)}\)
4. \(\left(\sqrt{25}\right)^{\left(\sqrt{25}\right)}\)
5. \(\left(\sqrt[3]{-27}\right)^{\left(\sqrt[3]{-27}\right)}\)
6. \(\left(\sqrt[5]{-32}\right)^{\left(\sqrt[5]{-32}\right)}\)

### Solutions:

1. **Expression 1: \(\left(\sqrt{4}\right)^{\left(\sqrt{4}\right)}\)**  
   - \(\sqrt{4} = 2\), so this becomes \(2^2 = 4\), which is a rational number.
   
2. **Expression 2: \(\left(\sqrt[3]{2}\right)^{\left(\sqrt[3]{2}\right)}\)**  
   - \(\sqrt[3]{2}\) is not an integer, and its exponent is also \(\sqrt[3]{2}\), a non-integer. The result is an irrational number.

3. **Expression 3: \(\left(\sqrt{4}\right)^{\left(\sqrt[2]{4}\right)}\)**  
   - \(\sqrt{4} = 2\) and \(\sqrt[2]{4} = 2\), so this is \(2^2 = 4\), which is rational.

4. **Expression 4: \(\left(\sqrt{25}\right)^{\left(\sqrt{25}\right)}\)**  
   - \(\sqrt{25} = 5\), so this becomes \(5^5\), which is a rational number.

5. **Expression 5: \(\left(\sqrt[3]{-27}\right)^{\left(\sqrt[3]{-27}\right)}\)**  
   - \(\sqrt[3]{-27} = -3\), so this becomes \((-3)^{-3} = -\frac{1}{27}\), which is a rational number.

6. **Expression 6: \(\left(\sqrt[5]{-32}\right)^{\left(\sqrt[5]{-32}\right)}\)**  
   - \(\sqrt[5]{-32} = -2\), so this becomes \((-2)^{-2} = \frac{1}{4}\), which is rational.

### Final Answer:
The correct expressions that represent rational numbers are:

1. \(\left(\sqrt{4}\right)^{\left(\sqrt{4}\right)}\)
3. \(\left(\sqrt{4}\right)^{\left(\sqrt[2]{4}\right)}\)
4. \(\left(\sqrt{25}\right)^{\left(\sqrt{25}\right)}\)
5. \(\left(\sqrt[3]{-27}\right)^{\left(\sqrt[3]{-27}\right)}\)
6. \(\left(\sqrt[5]{-32}\right)^{\left(\sqrt[5]{-32}\right)}\)

Do you want further details or have any questions?

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Here are some related questions for further exploration:

1. What defines a rational number?
2. How can you identify if an expression is rational?
3. What are the differences between rational and irrational numbers?
4. How does the order of operations affect the rationality of a number?
5. What are some examples of irrational numbers?
6. How do cube roots and square roots differ in terms of rationality?
7. What happens when you raise an irrational number to an irrational exponent?
8. How does negative exponents affect rational and irrational numbers?

**Tip:** Always simplify the expression before determining if it's rational or irrational.

Learn how to identify rational numbers using exponents and roots in mathematical expressions. This problem examines various expressions involving square roots and cube roots to determine if they result in rational numbers.

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