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.