Rationalize the denominator and simplify:

[tex]\[
\frac{2+\sqrt{6}}{\sqrt{3}}
\][/tex]

A. [tex]\(\frac{\sqrt{6}+3 \sqrt{2}}{3}\)[/tex]

B. [tex]\(\frac{2 \sqrt{3}+3 \sqrt{2}}{3}\)[/tex]

C. [tex]\(\frac{2 \sqrt{3}+\sqrt{6}}{3}\)[/tex]

D. [tex]\(\frac{2+3 \sqrt{2}}{3}\)[/tex]



Answer :

To rationalize the denominator and simplify the expression [tex]\(\frac{2 + \sqrt{6}}{\sqrt{3}}\)[/tex], we will follow a systematic step-by-step process.

### Step 1: Rationalize the denominator

To rationalize the denominator, we will multiply both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex], because [tex]\(\sqrt{3} \cdot \sqrt{3} = 3\)[/tex].

[tex]\[ \frac{2 + \sqrt{6}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{(2 + \sqrt{6}) \cdot \sqrt{3}}{3} \][/tex]

### Step 2: Simplify the numerator

We will now distribute [tex]\(\sqrt{3}\)[/tex] to each term in the numerator:

[tex]\[ (2 + \sqrt{6}) \cdot \sqrt{3} = 2\sqrt{3} + \sqrt{6} \cdot \sqrt{3} \][/tex]

### Step 3: Simplify the product [tex]\(\sqrt{6} \cdot \sqrt{3}\)[/tex]

Recall that [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\)[/tex]:

[tex]\[ \sqrt{6} \cdot \sqrt{3} = \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \][/tex]

### Step 4: Combine the terms in the numerator

Combine [tex]\(2\sqrt{3}\)[/tex] and [tex]\(3\sqrt{2}\)[/tex]:

[tex]\[ 2\sqrt{3} + 3\sqrt{2} \][/tex]

Thus, our fraction now looks like this:

[tex]\[ \frac{2\sqrt{3} + 3\sqrt{2}}{3} \][/tex]

### Step 5: Simplify the expression

Since the denominator is already a rational number (3), we can write the simplified result directly as:

[tex]\[ \frac{2\sqrt{3} + 3\sqrt{2}}{3} \][/tex]

### Final Answer

Therefore, the rationalized and simplified form of the expression is:

[tex]\[ \frac{2\sqrt{3} + 3\sqrt{2}}{3} \][/tex]

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