Let's go through the simplifications for each expression in turn:

a) $\sqrt{8} \times 2\sqrt{6}$

• Simplify $\sqrt{8}$: $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$
• Now the expression is $2\sqrt{2} \times 2\sqrt{6} = 4\sqrt{12}$
• Simplify $\sqrt{12}$: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$
• So the final result is: $4 \times 2\sqrt{3} = 8\sqrt{3}$

b) $-4\sqrt{5} \times 2\sqrt{13}$

• Multiply the constants: $-4 \times 2 = -8$
• Multiply the square roots: $\sqrt{5} \times \sqrt{13} = \sqrt{65}$
• So the final result is: $-8\sqrt{65}$

c) $\sqrt{2} \times \sqrt{6} \times 3\sqrt{3}$

• Multiply the square roots: $\sqrt{2} \times \sqrt{6} = \sqrt{12} = 2\sqrt{3}$
• Now the expression is: $2\sqrt{3} \times 3\sqrt{3} = 6\sqrt{9} = 6 \times 3 = 18$
• So the final result is: $18$

d) $3\sqrt{2} \times 5\sqrt{6}$

• Multiply the constants: $3 \times 5 = 15$
• Multiply the square roots: $\sqrt{2} \times \sqrt{6} = \sqrt{12} = 2\sqrt{3}$
• Now the expression is: $15 \times 2\sqrt{3} = 30\sqrt{3}$
• So the final result is: $30\sqrt{3}$

e) $2\sqrt{3} \times 3\sqrt{6}$

• Multiply the constants: $2 \times 3 = 6$
• Multiply the square roots: $\sqrt{3} \times \sqrt{6} = \sqrt{18} = 3\sqrt{2}$
• Now the expression is: $6 \times 3\sqrt{2} = 18\sqrt{2}$
• So the final result is: $18\sqrt{2}$

f) $5\sqrt{7} \times 2\sqrt{14}$

• Multiply the constants: $5 \times 2 = 10$
• Multiply the square roots: $\sqrt{7} \times \sqrt{14} = \sqrt{98} = 7\sqrt{2}$
• Now the expression is: $10 \times 7\sqrt{2} = 70\sqrt{2}$
• So the final result is: $70\sqrt{2}$

g) $\sqrt{6} \times \sqrt{2}$

• Multiply the square roots: $\sqrt{6} \times \sqrt{2} = \sqrt{12} = 2\sqrt{3}$
• So the final result is: $2\sqrt{3}$

h) $\sqrt{2} \times 3\sqrt{3}$

• Multiply the constants: $1 \times 3 = 3$
• Multiply the square roots: $\sqrt{2} \times \sqrt{3} = \sqrt{6}$
• So the final result is: $3\sqrt{6}$

i) $5\sqrt{3} \times 2\sqrt{2}$

• Multiply the constants: $5 \times 2 = 10$
• Multiply the square roots: $\sqrt{3} \times \sqrt{2} = \sqrt{6}$
• So the final result is: $10\sqrt{6}$

j) $\sqrt{3} \times \sqrt{7}$

• Multiply the square roots: $\sqrt{3} \times \sqrt{7} = \sqrt{21}$
• So the final result is: $\sqrt{21}$

Would you like more detailed explanations for any step?

Related Questions:

1. How do you simplify expressions involving square roots in general?
2. Can you explain the difference between rational and irrational numbers?
3. How do you rationalize the denominator in a fraction with a square root?
4. What is the process for simplifying a product of two square roots?
5. How do you simplify square roots with larger numbers?

Tip:

When multiplying square roots, remember to multiply the numbers inside the square roots first, and simplify only after.