Let's solve these square root problems step-by-step.

Problem #10

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

1. Separate into Square Roots: Break down each term: $\sqrt{9} \times \sqrt{x^2} \times \sqrt{y^5}$

2. Simplify Each Term:

   • $\sqrt{9} = 3$
   • $\sqrt{x^2} = x$ (since $x^2$ is a perfect square)
   • For $\sqrt{y^5}$, separate $y^5$ as $y^4 \times y$: $\sqrt{y^5} = \sqrt{y^4 \times y} = \sqrt{y^4} \times \sqrt{y} = y^2 \sqrt{y}$

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

So, the answer for #10 is: $3xy^2 \sqrt{y}$

Problem #12

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

1. Separate into Square Roots: $\sqrt{32} \times \sqrt{a^4} \times \sqrt{b^7}$

2. Simplify Each Term:

   • $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$
   • $\sqrt{a^4} = a^2$ (since $a^4$ is a perfect square)
   • For $\sqrt{b^7}$, separate $b^7$ as $b^6 \times b$: $\sqrt{b^7} = \sqrt{b^6 \times b} = \sqrt{b^6} \times \sqrt{b} = b^3 \sqrt{b}$

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

So, the answer for #12 is: $4a^2b^3 \sqrt{2b}$

Would you like more details on any part of the solution, or do you have any specific questions?

Related Questions to Explore

1. How do you simplify square roots with variables?
2. What is the difference between perfect squares and non-perfect squares?
3. How do you handle square roots when there are fractional exponents?
4. How can you identify terms that can be simplified inside a square root?
5. Why is it helpful to break down terms into their factors?

Tip:

When simplifying square roots with variables, separate each component (constants and variables) and look for any perfect squares. This makes it easier to simplify each part individually.