https://file.aiquickdraw.com/mathbot/file/17251015258789otu7uar.jpg

Let's solve the problems presented in the image. The task is to simplify the expressions involving square roots. I'll go through each problem step by step.

1. **\( 3(5\sqrt{2}) \)**

   Simplification:
   \[
   3 \times 5\sqrt{2} = 15\sqrt{2}
   \]

2. **\( 2(3\sqrt{15}) \)**

   Simplification:
   \[
   2 \times 3\sqrt{15} = 6\sqrt{15}
   \]

3. **\( 3(\sqrt{3} - 4\sqrt{5}) \)**

   Distribute the 3:
   \[
   3\sqrt{3} - 12\sqrt{5}
   \]

4. **\( \sqrt{3}(5\sqrt{6} - 4\sqrt{3}) \)**

   Distribute \( \sqrt{3} \):
   \[
   5\sqrt{18} - 4\sqrt{9}
   \]
   Simplify further:
   \[
   5 \times 3\sqrt{2} - 4 \times 3 = 15\sqrt{2} - 12
   \]

5. **\( (1 + \sqrt{3})(3 + \sqrt{2}) \)**

   Use FOIL (First, Outside, Inside, Last):
   \[
   1 \times 3 + 1 \times \sqrt{2} + \sqrt{3} \times 3 + \sqrt{3} \times \sqrt{2}
   \]
   Simplify:
   \[
   3 + \sqrt{2} + 3\sqrt{3} + \sqrt{6}
   \]

6. **\( (\sqrt{2} + \sqrt{12})(\sqrt{18} + 3\sqrt{3}) \)**

   Simplify square roots:
   \[
   (\sqrt{2} + 2\sqrt{3})(3\sqrt{2} + 3\sqrt{3})
   \]
   Then use FOIL:
   \[
   \sqrt{2} \times 3\sqrt{2} + \sqrt{2} \times 3\sqrt{3} + 2\sqrt{3} \times 3\sqrt{2} + 2\sqrt{3} \times 3\sqrt{3}
   \]
   Simplify:
   \[
   6 + 3\sqrt{6} + 6\sqrt{6} + 6\sqrt{9} = 6 + 9\sqrt{6} + 18
   \]
   \[
   = 24 + 9\sqrt{6}
   \]

7. **\( (\sqrt{3} - 2\sqrt{5})(\sqrt{3} - \sqrt{5}) \)**

   Use FOIL:
   \[
   \sqrt{3} \times \sqrt{3} - \sqrt{3} \times \sqrt{5} - 2\sqrt{5} \times \sqrt{3} + 2\sqrt{5} \times \sqrt{5}
   \]
   Simplify:
   \[
   3 - \sqrt{15} - 2\sqrt{15} + 10
   \]
   \[
   = 13 - 3\sqrt{15}
   \]

8. **\( (\sqrt{3} - \sqrt{5})^2 \)**

   Use the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
   \[
   3 - 2\sqrt{15} + 5
   \]
   \[
   = 8 - 2\sqrt{15}
   \]

9. **\( (2\sqrt{3} + \sqrt{2})(2\sqrt{3} - \sqrt{2}) \)**

   Use the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):
   \[
   (2\sqrt{3})^2 - (\sqrt{2})^2 = 4 \times 3 - 2 = 12 - 2 = 10
   \]

10. **\( (1 - \sqrt{6})(4 + \sqrt{2}) \)**

    Use FOIL:
    \[
    1 \times 4 + 1 \times \sqrt{2} - \sqrt{6} \times 4 - \sqrt{6} \times \sqrt{2}
    \]
    Simplify:
    \[
    4 + \sqrt{2} - 4\sqrt{6} - \sqrt{12} = 4 + \sqrt{2} - 4\sqrt{6} - 2\sqrt{3}
    \]

---

Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How can we generalize the FOIL method for more than two binomials?
2. What are some common errors to avoid when simplifying radical expressions?
3. How does the distributive property apply to the multiplication of binomials with radicals?
4. Can you derive the formula for the square of a binomial and apply it to more complex expressions?
5. What is the significance of rationalizing the denominator in expressions with radicals?

**Tip:** When dealing with square roots, always try to simplify them as much as possible before proceeding with other operations.

Explore how to simplify radical expressions step-by-step with detailed solutions. Covering concepts like radical expressions, FOIL method, difference of squares, and distributive property. Suitable for high school students and anyone interested in algebraic simplification.

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