https://file.aiquickdraw.com/mathbot/file/1726759427142hksvnegd.jpg

Let's break down the expressions to determine which ones represent irrational numbers.

### Recall:
- A **rational number** is any number that can be expressed as a ratio of two integers (e.g., fractions, integers).
- An **irrational number** is a number that **cannot** be written as a simple fraction, like \( \pi \) or \( \sqrt{2} \).

### Checking the expressions:

1. **\( (\sqrt{5})^2 + (\sqrt{17})^3 \)**:
   - \( (\sqrt{5})^2 = 5 \) (rational number).
   - \( (\sqrt{17})^3 = \sqrt{17} \times \sqrt{17} \times \sqrt{17} = 17\sqrt{17} \) (irrational number).
   - Since one part is irrational, **this expression is irrational**.

2. **\( (\sqrt{4})^3 (\sqrt{0.81}) \)**:
   - \( \sqrt{4} = 2 \), and \( 2^3 = 8 \) (rational number).
   - \( \sqrt{0.81} = 0.9 \) (rational number).
   - The product of rational numbers is rational. **This expression is rational**.

3. **\( (\sqrt{16})^3 + (\sqrt{12})^2 \)**:
   - \( \sqrt{16} = 4 \), and \( 4^3 = 64 \) (rational number).
   - \( (\sqrt{12})^2 = 12 \) (rational number).
   - Sum of rational numbers is rational. **This expression is rational**.

4. **\( (\sqrt{3})^3 (\sqrt{36})^2 \)**:
   - \( \sqrt{3}^3 = \sqrt{3} \times \sqrt{3} \times \sqrt{3} = 3\sqrt{3} \) (irrational number).
   - \( (\sqrt{36})^2 = 36 \) (rational number).
   - The product of an irrational number and a rational number is irrational. **This expression is irrational**.

5. **\( (\sqrt{0.09}) + (\sqrt{21})^2 \)**:
   - \( \sqrt{0.09} = 0.3 \) (rational number).
   - \( (\sqrt{21})^2 = 21 \) (rational number).
   - Sum of rational numbers is rational. **This expression is rational**.

### Conclusion:
The two expressions that represent irrational numbers are:
1. \( (\sqrt{5})^2 + (\sqrt{17})^3 \)
2. \( (\sqrt{3})^3 (\sqrt{36})^2 \)

Would you like any further clarification?

Here are some related questions to expand on this:
1. What is the definition of a rational and irrational number?
2. How do you simplify expressions involving square roots?
3. Can a product of an irrational and a rational number ever be rational?
4. How do you distinguish between rational and irrational numbers without calculating?
5. Why is \( \sqrt{2} \) considered an irrational number?

**Tip:** When working with square roots, irrational numbers often arise when the number inside the root is not a perfect square.

Which two expressions represent irrational numbers?

Learn how to identify irrational numbers in algebraic expressions involving square roots and exponents. This problem helps to distinguish between rational and irrational numbers in expressions, making it suitable for students in Grades 8-10.

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