Find the value of `(2 + sqrt(3))/(2 - sqrt(3))`.

Answer

To find the value of \(\frac{2 + \sqrt{3}}{2 - \sqrt{3}}\), we will rationalize the denominator. Here’s a step-by-step solution:

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

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(2 + \sqrt{3}\):
\[ \frac{(2 + \sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]

### Step 3: Expand the numerator
Now we expand the numerator using the formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[ (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Step 4: Expand the denominator
Next, we expand the denominator using the difference of squares formula \(a^2 - b^2\):
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 5: Write the final expression
Now we can write the expression as:
\[ \frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3} \]

### Final Answer
Thus, the value of \(\frac{2 + \sqrt{3}}{2 - \sqrt{3}}\) is:
\[ \boxed{7 + 4\sqrt{3}} \]