Let's go through each problem step-by-step. I’ll make it as simple and quick as possible.

Problem #10:

Simplify $\sqrt{9x^2y^5}$

1. Separate the square root: $\sqrt{9x^2y^5} = \sqrt{9} \cdot \sqrt{x^2} \cdot \sqrt{y^5}$

2. Simplify each term:

   • $\sqrt{9} = 3$
   • $\sqrt{x^2} = x$
   • For $\sqrt{y^5}$, write $y^5$ as $y^4 \cdot y$: $\sqrt{y^5} = \sqrt{y^4} \cdot \sqrt{y} = y^2 \sqrt{y}$

3. Combine everything: $\sqrt{9x^2y^5} = 3x \cdot y^2 \sqrt{y} = 3x y^2 \sqrt{y}$

Answer: $3x y^2 \sqrt{y}$

Problem #12:

Simplify $\sqrt{32a^4b^7}$

1. Separate terms: $\sqrt{32a^4b^7} = \sqrt{32} \cdot \sqrt{a^4} \cdot \sqrt{b^7}$

2. Simplify each term:

   • $\sqrt{32} = 4\sqrt{2}$ because $32 = 16 \times 2$ and $\sqrt{16} = 4$.
   • $\sqrt{a^4} = a^2$
   • For $\sqrt{b^7}$, write $b^7$ as $b^6 \cdot b$: $\sqrt{b^7} = \sqrt{b^6} \cdot \sqrt{b} = b^3 \sqrt{b}$

3. Combine everything: $\sqrt{32a^4b^7} = 4\sqrt{2} \cdot a^2 \cdot b^3 \sqrt{b} = 4a^2b^3 \sqrt{2b}$

Answer: $4a^2b^3 \sqrt{2b}$

Problem #16:

Simplify $\sqrt[3]{16a^4b^5}$

1. Separate terms: [ \sqrt[3]{16a^4b^5} = \sqrt[3]{16} \cdot \sqrt[3]{a^4} \cdot \sqrt[3]{b^5} ]

2. Simplify each term:

   • $\sqrt[3]{16} = 2 \sqrt[3]{2}$ because $16 = 8 \times 2$ and $\sqrt[3]{8} = 2$.
   • $\sqrt[3]{a^4} = a^{4/3}$
   • $\sqrt[3]{b^5} = b^{5/3}$

3. Combine everything: [ \sqrt[3]{16a^4b^5} = 2a^{4/3}b^{5/3} \sqrt[3]{2} ]

Answer: $2a^{4/3}b^{5/3} \sqrt[3]{2}$

Clicker #2:

Simplify $\sqrt{32x^2y}$

1. Separate terms: $\sqrt{32x^2y} = \sqrt{32} \cdot \sqrt{x^2} \cdot \sqrt{y}$

2. Simplify each term:

   • $\sqrt{32} = 4\sqrt{2}$
   • $\sqrt{x^2} = x$
   • $\sqrt{y} = \sqrt{y}$

3. Combine everything: $\sqrt{32x^2y} = 4x\sqrt{2y}$

Answer: $\text{c) } 4x \sqrt{2y}$

Would you like more details on any part?

Follow-up Questions:

1. How do you handle variables with odd exponents under a square root?
2. What is the difference between square roots and cube roots in simplifying expressions?
3. How do you identify perfect squares or cubes within an expression?
4. What are some tips for simplifying expressions with multiple terms under a root?
5. How does fractional exponent notation relate to roots?

Tip: When simplifying roots, always look to separate constants and variables into terms that are perfect squares (or cubes, etc.) to make simplification easier.