Rationalize the denominator `(6)/(2sqrt(3)+sqrt(6))`

Answer

To rationalize the denominator of the expression \(\frac{6}{2\sqrt{3} + \sqrt{6}}\), we will follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{6}{2\sqrt{3} + \sqrt{6}} \]

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2\sqrt{3} + \sqrt{6}\) is \(2\sqrt{3} - \sqrt{6}\).

So, we multiply:
\[ \frac{6}{2\sqrt{3} + \sqrt{6}} \cdot \frac{2\sqrt{3} - \sqrt{6}}{2\sqrt{3} - \sqrt{6}} \]

### Step 3: Apply the multiplication
Now, we perform the multiplication in the numerator and the denominator.

**Numerator:**
\[ 6 \cdot (2\sqrt{3} - \sqrt{6}) = 12\sqrt{3} - 6\sqrt{6} \]

**Denominator:**
Using the identity \( (a + b)(a - b) = a^2 - b^2 \):
\[ (2\sqrt{3})^2 - (\sqrt{6})^2 = 4 \cdot 3 - 6 = 12 - 6 = 6 \]

### Step 4: Write the new expression
Now we can write the expression as:
\[ \frac{12\sqrt{3} - 6\sqrt{6}}{6} \]

### Step 5: Simplify the expression
We can simplify the expression by dividing each term in the numerator by 6:
\[ \frac{12\sqrt{3}}{6} - \frac{6\sqrt{6}}{6} = 2\sqrt{3} - \sqrt{6} \]

### Final Answer:
Thus, the rationalized form of the expression \(\frac{6}{2\sqrt{3} + \sqrt{6}}\) is:
\[ 2\sqrt{3} - \sqrt{6} \]

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