Let's solve each problem step-by-step with explanations. We'll simplify the terms in each equation and combine like terms.

1. $3\sqrt{2x} + 5\sqrt{2x}$

• Both terms have the same radical ($\sqrt{2x}$), so we can add the coefficients: $3 + 5 = 8$
• Final result: $8\sqrt{2x}$

2. $4\sqrt{3y} - 7\sqrt{3y}$

• Both terms have the same radical ($\sqrt{3y}$), so subtract the coefficients: $4 - 7 = -3$
• Final result: $-3\sqrt{3y}$

3. $6\sqrt{5z} + 2\sqrt{5z}$

• Both terms have the same radical ($\sqrt{5z}$), so add the coefficients: $6 + 2 = 8$
• Final result: $8\sqrt{5z}$

4. $8\sqrt{6a} - 3\sqrt{6a}$

• Both terms have the same radical ($\sqrt{6a}$), so subtract the coefficients: $8 - 3 = 5$
• Final result: $5\sqrt{6a}$

5. $5\sqrt{7b} + 4\sqrt{7b}$

• Both terms have the same radical ($\sqrt{7b}$), so add the coefficients: $5 + 4 = 9$
• Final result: $9\sqrt{7b}$

6. $2\sqrt{10c} - 6\sqrt{5c}$

• These radicals ($\sqrt{10c}$ and $\sqrt{5c}$) are not like terms because their radicands are different. Therefore, this expression cannot be simplified further: $2\sqrt{10c} - 6\sqrt{5c}$

7. $3\sqrt{2x} + 4\sqrt{3x}$

• These radicals ($\sqrt{2x}$ and $\sqrt{3x}$) are not like terms because their radicands are different. Therefore, this expression cannot be simplified further: $3\sqrt{2x} + 4\sqrt{3x}$

8. $7\sqrt{3y} - 2\sqrt{12y}$

• Simplify $\sqrt{12y}$ first: $\sqrt{12y} = \sqrt{4 \cdot 3y} = 2\sqrt{3y}$
• Substitute back into the equation: $7\sqrt{3y} - 2(2\sqrt{3y}) = 7\sqrt{3y} - 4\sqrt{3y}$
• Combine like terms: $7 - 4 = 3$
• Final result: $3\sqrt{3y}$

9. $9\sqrt{4z} + 5\sqrt{2z}$

• Simplify $\sqrt{4z}$ first: $\sqrt{4z} = 2\sqrt{z}$
• Substitute back into the equation: $9(2\sqrt{z}) + 5\sqrt{2z} = 18\sqrt{z} + 5\sqrt{2z}$
• These terms are not like terms because their radicands are different. Therefore, the simplified result is: $18\sqrt{z} + 5\sqrt{2z}$

10. $6\sqrt{5a} - 8\sqrt{10a}$

• These radicals ($\sqrt{5a}$ and $\sqrt{10a}$) are not like terms because their radicands are different. Therefore, this expression cannot be simplified further: $6\sqrt{5a} - 8\sqrt{10a}$

If you'd like, I can explain further details on any specific problem or expand on related topics!

5 Questions to Expand:

1. How can we simplify radicals with different radicands in more complex cases?
2. What are the rules for combining coefficients with like radicals?
3. Can you explain how to factor out square roots in expressions?
4. How do these simplifications relate to solving equations involving radicals?
5. Could you demonstrate additional examples of simplifying radicals?

Tip: Always check if the radicand can be factored into a perfect square to simplify expressions more effectively.