Rationlise the denominator and simplify:
` (3)/( 4- sqrt3) `

Answer

To rationalize the denominator and simplify the expression \(\frac{3}{4 - \sqrt{3}}\), we will follow these steps:

### Step 1: 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 \(4 - \sqrt{3}\) is \(4 + \sqrt{3}\).

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

### Step 2: Apply the Multiplication
Now, we will perform the multiplication in both the numerator and the denominator.

**Numerator:**
\[ 3 \times (4 + \sqrt{3}) = 12 + 3\sqrt{3} \]

**Denominator:**
Using the formula \(a^2 - b^2\) where \(a = 4\) and \(b = \sqrt{3}\):
\[ (4 - \sqrt{3})(4 + \sqrt{3}) = 4^2 - (\sqrt{3})^2 = 16 - 3 = 13 \]

### Step 3: Combine the Results
Now we can combine the results from the numerator and the denominator:

\[ \frac{12 + 3\sqrt{3}}{13} \]

### Final Expression
Thus, the rationalized and simplified form of \(\frac{3}{4 - \sqrt{3}}\) is:

\[ \frac{12 + 3\sqrt{3}}{13} \]

---