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]
