3√2÷√6+√3+4√3÷√6+√2−(√2÷√6−√2

Let's break down the expression step by step:

Given expression:

\[
\frac{3\sqrt{2}}{\sqrt{6}} + \sqrt{3} + \frac{4\sqrt{3}}{\sqrt{6}} + \sqrt{2} - \left(\frac{\sqrt{2}}{\sqrt{6}} - \sqrt{2}\right)
\]

### Step 1: Simplify each term

1. **Simplify \(\frac{3\sqrt{2}}{\sqrt{6}}\):**

\[
\frac{3\sqrt{2}}{\sqrt{6}} = \frac{3\sqrt{2}}{\sqrt{2 \times 3}} = \frac{3\sqrt{2}}{\sqrt{2} \times \sqrt{3}} = \frac{3\sqrt{2}}{\sqrt{2}\sqrt{3}} = \frac{3}{\sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3}
\]

2. **Simplify \(\frac{4\sqrt{3}}{\sqrt{6}}\):**

\[
\frac{4\sqrt{3}}{\sqrt{6}} = \frac{4\sqrt{3}}{\sqrt{2} \times \sqrt{3}} = \frac{4}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}
\]

3. **Simplify \(\frac{\sqrt{2}}{\sqrt{6}}\):**

\[
\frac{\sqrt{2}}{\sqrt{6}} = \frac{\sqrt{2}}{\sqrt{2 \times 3}} = \frac{\sqrt{2}}{\sqrt{2}\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]

### Step 2: Substitute simplified terms back into the expression

The expression now becomes:

\[
\sqrt{3} + \sqrt{3} + 2\sqrt{2} + \sqrt{2} - \left(\frac{\sqrt{3}}{3} - \sqrt{2}\right)
\]

### Step 3: Combine like terms

Combine the like terms:

1. Combine \(\sqrt{3} + \sqrt{3}\):

\[
\sqrt{3} + \sqrt{3} = 2\sqrt{3}
\]

2. Combine \(2\sqrt{2} + \sqrt{2}\):

\[
2\sqrt{2} + \sqrt{2} = 3\sqrt{2}
\]

The expression is now:

\[
2\sqrt{3} + 3\sqrt{2} - \left(\frac{\sqrt{3}}{3} - \sqrt{2}\right)
\]

### Step 4: Distribute the negative sign and simplify

Distribute the negative sign inside the parentheses:

\[
2\sqrt{3} + 3\sqrt{2} - \frac{\sqrt{3}}{3} + \sqrt{2}
\]

Now combine the like terms again:

1. Combine \(3\sqrt{2} + \sqrt{2}\):

\[
3\sqrt{2} + \sqrt{2} = 4\sqrt{2}
\]

The expression becomes:

\[
2\sqrt{3} - \frac{\sqrt{3}}{3} + 4\sqrt{2}
\]

### Step 5: Combine the \(\sqrt{3}\) terms

To combine \(2\sqrt{3}\) and \(-\frac{\sqrt{3}}{3}\), express them with a common denominator:

\[
2\sqrt{3} = \frac{6\sqrt{3}}{3}
\]

So:

\[
\frac{6\sqrt{3}}{3} - \frac{\sqrt{3}}{3} = \frac{5\sqrt{3}}{3}
\]

### Final Expression

\[
\frac{5\sqrt{3}}{3} + 4\sqrt{2}
\]

This is the simplified form of the original expression.

Would you like to explore this further or ask questions?

### Relative Questions:
1. How can we rationalize denominators in expressions like this?
2. Can you simplify \(\frac{\sqrt{5}}{\sqrt{10}}\) similarly?
3. How do we add or subtract radicals with different radicands?
4. What is the geometric interpretation of \(\sqrt{2}\) and \(\sqrt{3}\)?
5. How does this process differ when dealing with cube roots instead of square roots?

### Tip:
When simplifying radical expressions, always look for opportunities to combine like terms and rationalize denominators early in the process.

Learn how to simplify the radical expression 3√2÷√6+√3+4√3÷√6+√2−(√2÷√6−√2) step-by-step. This problem involves square roots, rationalizing denominators, and combining like terms. Suitable for high school students (Grades 9-12).

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