3√2÷√6+√3+4√3÷√6++√2−(√2÷√6−√2

Let's break down the expression step by step:

The given expression is:
\[
3\sqrt{2} \div \sqrt{6} + \sqrt{3} + 4\sqrt{3} \div \sqrt{6} + \sqrt{2} - \left(\frac{\sqrt{2}}{\sqrt{6}} - \sqrt{2}\right)
\]

### Step 1: Simplify each square root division
1. \(\frac{\sqrt{2}}{\sqrt{6}} = \frac{\sqrt{2}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{12}}{6} = \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3}\)
2. \(3\sqrt{2} \div \sqrt{6} = 3 \times \frac{\sqrt{2}}{\sqrt{6}} = 3 \times \frac{\sqrt{3}}{3} = \sqrt{3}\)
3. \(4\sqrt{3} \div \sqrt{6} = 4 \times \frac{\sqrt{3}}{\sqrt{6}} = 4 \times \frac{\sqrt{3}}{\sqrt{6}} = 4 \times \frac{\sqrt{6}}{6} = \frac{4\sqrt{2}}{3}\)

### Step 2: Substitute back into the expression
Now, substituting the simplified forms into the original expression:
\[
\sqrt{3} + \sqrt{3} + \frac{4\sqrt{2}}{3} + \sqrt{2} - \left(\frac{\sqrt{3}}{3} - \sqrt{2}\right)
\]

### Step 3: Combine like terms
1. Combine the \(\sqrt{3}\) terms:
\[
\sqrt{3} + \sqrt{3} = 2\sqrt{3}
\]

2. Combine \(\sqrt{2}\) terms:
\[
\sqrt{2} + \sqrt{2} = 2\sqrt{2}
\]

3. Consider the subtraction part:
\[
\text{Expression becomes: } 2\sqrt{3} + \frac{4\sqrt{2}}{3} - \frac{\sqrt{3}}{3} + 2\sqrt{2}
\]

### Step 4: Simplify the entire expression
Combine \(\sqrt{3}\) terms:
\[
2\sqrt{3} - \frac{\sqrt{3}}{3} = \frac{6\sqrt{3}}{3} - \frac{\sqrt{3}}{3} = \frac{5\sqrt{3}}{3}
\]
Now the expression is:
\[
\frac{5\sqrt{3}}{3} + \frac{4\sqrt{2}}{3} + 2\sqrt{2}
\]

Finally, combining the \(\sqrt{2}\) terms:
\[
\frac{5\sqrt{3}}{3} + \frac{4\sqrt{2}}{3} + \frac{6\sqrt{2}}{3} = \frac{5\sqrt{3}}{3} + \frac{10\sqrt{2}}{3}
\]

### Final Simplified Expression:
\[
\frac{5\sqrt{3} + 10\sqrt{2}}{3}
\]

This is the simplified form of the given expression.

Do you want details or have any questions?

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Here are 5 related questions for further exploration:
1. How do you simplify square root expressions involving division?
2. What are some strategies to combine like terms when square roots are involved?
3. How do you rationalize the denominator in square root expressions?
4. How would the approach change if one of the terms was a fraction instead of a square root?
5. Can you demonstrate how to simplify an expression with mixed radical terms?

**Tip:** When simplifying expressions involving radicals, always look for common factors and opportunities to rationalize the denominator for a more straightforward solution.

Learn how to simplify the square root expression 3√2÷√6+√3+4√3÷√6++√2−(√2÷√6−√2) step-by-step. Understand division of square roots, combining like terms with radicals, and simplification techniques. Suitable for high school students studying algebra and radicals.

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