The image contains five different mathematical expressions involving square roots. Let's evaluate each expression step by step:

a. $4 \sqrt{3} \times 2 \sqrt{3}$

• First, multiply the numerical coefficients: $4 \times 2 = 8$.
• Then, multiply the square roots: $\sqrt{3} \times \sqrt{3} = 3$.
• Combine the results: $8 \times 3 = 24$.

Result: $24$

b. $\sqrt{2} \left( \sqrt{3} + \sqrt{5} \right)$

• Distribute $\sqrt{2}$ to both terms inside the parentheses:
  • $\sqrt{2} \times \sqrt{3} = \sqrt{6}$
  • $\sqrt{2} \times \sqrt{5} = \sqrt{10}$
• The expression simplifies to: $\sqrt{6} + \sqrt{10}$.

Result: $\sqrt{6} + \sqrt{10}$

c. $\sqrt{2} \times \left( \sqrt{3} \times \sqrt{5} \right)$

• First, multiply the square roots inside the parentheses: $\sqrt{3} \times \sqrt{5} = \sqrt{15}$.
• Then multiply by $\sqrt{2}$: $\sqrt{2} \times \sqrt{15} = \sqrt{30}$.

Result: $\sqrt{30}$

d. $2 \sqrt{3} \left( \sqrt{2} - \sqrt{5} \right)$

• Distribute $2 \sqrt{3}$ to both terms inside the parentheses:
  • $2 \sqrt{3} \times \sqrt{2} = 2 \sqrt{6}$
  • $2 \sqrt{3} \times -\sqrt{5} = -2 \sqrt{15}$
• The expression simplifies to: $2 \sqrt{6} - 2 \sqrt{15}$.

Result: $2 \sqrt{6} - 2 \sqrt{15}$

e. $2 \sqrt{3} \times 4 \sqrt{2} \times 3 \sqrt{5}$

• First, multiply the numerical coefficients: $2 \times 4 \times 3 = 24$.
• Then, multiply the square roots:
  • $\sqrt{3} \times \sqrt{2} = \sqrt{6}$
  • $\sqrt{6} \times \sqrt{5} = \sqrt{30}$
• Combine the results: $24 \times \sqrt{30}$.

Result: $24 \sqrt{30}$

Summary of Results:

• a. $24$
• b. $\sqrt{6} + \sqrt{10}$
• c. $\sqrt{30}$
• d. $2 \sqrt{6} - 2 \sqrt{15}$
• e. $24 \sqrt{30}$

Would you like more details on any of these solutions, or do you have any questions?

Related Questions:

1. How do you simplify expressions involving multiple square roots?
2. What are some methods for multiplying and adding square roots?
3. How does the distributive property apply when working with square roots?
4. Can you simplify the result of multiplying $\sqrt{a} \times \sqrt{b}$?
5. What is the significance of rationalizing the denominator in square root expressions?

Tip:

When multiplying square roots, remember that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.