https://file.aiquickdraw.com/mathbot/file/1727593913731337fmut8.jpg

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: \(\sqrt{21}\)

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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.

Simplify the following expressions involving square roots:
 a) √8 × 2√6 
 b) -4√5 × 2√13 
 c) √2 × √6 × 3√3 
 d) 3√2 × 5√6 
 e) 2√3 × 3√6 
 f) 5√7 × 2√14 
 g) √6 × √2 
 h) √2 × 3√3 
 i) 5√3 × 2√2 
 j) √3 × √7

This page provides step-by-step solutions to simplifying expressions with square roots, covering problems like √8 × 2√6 and √3 × √7. Ideal for students in Grades 8-10 learning radical simplifications.

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