If a = `(1)/(2 - sqrt(3)) , b = (1)/(2 + sqrt(3))`, find the value of `((a + b)/(a -b))^(2)`.

Options

A. `3/4`
B. `2/3`
C. `4/3`
D. `4/5`

Answer

To solve the problem, we need to find the value of \(\left(\frac{a + b}{a - b}\right)^{2}\) given that \(a = \frac{1}{2 - \sqrt{3}}\) and \(b = \frac{1}{2 + \sqrt{3}}\).

### Step 1: Rationalize \(a\)

To rationalize \(a\):
\[ a = \frac{1}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})} \]
Using the difference of squares:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus, we have:
\[ a = 2 + \sqrt{3} \]

### Step 2: Rationalize \(b\)

Next, we rationalize \(b\):
\[ b = \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} \]
Again using the difference of squares:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 1 \]
Thus, we have:
\[ b = 2 - \sqrt{3} \]

### Step 3: Calculate \(a + b\)

Now we calculate \(a + b\):
\[ a + b = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \]

### Step 4: Calculate \(a - b\)

Next, we calculate \(a - b\):
\[ a - b = (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2 + \sqrt{3} - 2 + \sqrt{3} = 2\sqrt{3} \]

### Step 5: Calculate \(\frac{a + b}{a - b}\)

Now we can find \(\frac{a + b}{a - b}\):
\[ \frac{a + b}{a - b} = \frac{4}{2\sqrt{3}} = \frac{4}{2\sqrt{3}} = \frac{2}{\sqrt{3}} \]

### Step 6: Calculate \(\left(\frac{a + b}{a - b}\right)^{2}\)

Finally, we calculate:
\[ \left(\frac{a + b}{a - b}\right)^{2} = \left(\frac{2}{\sqrt{3}}\right)^{2} = \frac{4}{3} \]

Thus, the final answer is:
\[ \boxed{\frac{4}{3}} \]