Let’s work through the questions one by one with explanations:

1. $\sqrt{x} \cdot \sqrt{y}$

Using the property of radicals:
$\sqrt{x} \cdot \sqrt{y} = \sqrt{xy}$
Answer: $c. \sqrt{xy}$

2. $\frac{\sqrt{x}}{\sqrt{y}}$

Using the property of radicals:
$\frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}}$
Answer: $a. \sqrt{\frac{x}{y}}$

3. $\sqrt{48}$

Break 48 into prime factors:
$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$
Answer: $c. 4\sqrt{3}$

4. $\sqrt[3]{27}$

Cube root of 27 is:
$\sqrt[3]{27} = 3$
Answer: $c. 3$

5. Simplify $\sqrt{4}$

Square root of 4 is:
$\sqrt{4} = 2$
Answer: $d. 2$

6. Simplify $\sqrt{72x^5y^3}$

Simplify the radical expression:
$\sqrt{72x^5y^3} = \sqrt{36 \cdot 2 \cdot x^4 \cdot x \cdot y^2 \cdot y} = 6x^2y\sqrt{2xy}$
Answer: $c. 6x^2y\sqrt{2xy}$

7. Product $(3\sqrt{2})(4\sqrt{8})$

Simplify each term:
$3\sqrt{2} \cdot 4\sqrt{8} = 12\sqrt{16} = 12 \cdot 4 = 48$
Answer: $b. 48$

8. Examples of similar radicals

Radicals must have the same radicand to be similar.
In $\sqrt{7}, 4\sqrt{7}, -1\sqrt{7}$, all have $\sqrt{7}$.
But in $\sqrt{xy}, 4\sqrt{xy}, -1\sqrt{xy}$, not all terms are the same.
Answer: $d. \sqrt{xy}, 4\sqrt{xy}, -1\sqrt{xy}$

9. Sum $\sqrt{xy} + 4\sqrt{xy}$

Combine like terms:
$\sqrt{xy} + 4\sqrt{xy} = 5\sqrt{xy}$
Answer: $c. 5\sqrt{xy}$

10. Sum $5\sqrt{6} + 4\sqrt{24}$

Simplify $4\sqrt{24}$:
$4\sqrt{24} = 4\sqrt{4 \cdot 6} = 4 \cdot 2\sqrt{6} = 8\sqrt{6}$
Add the radicals:
$5\sqrt{6} + 8\sqrt{6} = 13\sqrt{6}$
Answer: $a. 13\sqrt{6}$

Do you want further details or have any specific questions about these solutions?

Five Follow-Up Questions:

1. Can you explain the general properties of radicals?
2. How do you simplify expressions involving cube roots?
3. What are some common errors when simplifying radical expressions?
4. How do you identify similar radicals in expressions?
5. What is the difference between simplifying and rationalizing radicals?

Tip:

Always break the radicand into prime factors when simplifying radicals. This ensures accuracy