If `(3+2sqrt(3))/(3-sqrt(3))=a+sqrt(3)b` , then the value of `sqrt(a+b)` , where a and b are rational numbers, is

Options

A. `5`
B. `8`
C. `2`
D. `16`

Answer

To solve the problem \(\frac{3 + 2\sqrt{3}}{3 - \sqrt{3}} = a + \sqrt{3}b\), we will follow these steps:

### Step 1: Rationalize the denominator
To simplify the expression, we will multiply the numerator and denominator by the conjugate of the denominator, which is \(3 + \sqrt{3}\).

\[ \frac{3 + 2\sqrt{3}}{3 - \sqrt{3}} \cdot \frac{3 + \sqrt{3}}{3 + \sqrt{3}} = \frac{(3 + 2\sqrt{3})(3 + \sqrt{3})}{(3 - \sqrt{3})(3 + \sqrt{3})} \]

### Step 2: Simplify the denominator
Using the difference of squares formula \(a^2 - b^2\):

\[ (3 - \sqrt{3})(3 + \sqrt{3}) = 3^2 - (\sqrt{3})^2 = 9 - 3 = 6 \]

### Step 3: Expand the numerator
Now we will expand the numerator:

\[ (3 + 2\sqrt{3})(3 + \sqrt{3}) = 3 \cdot 3 + 3 \cdot \sqrt{3} + 2\sqrt{3} \cdot 3 + 2\sqrt{3} \cdot \sqrt{3} \]
\[ = 9 + 3\sqrt{3} + 6\sqrt{3} + 2 \cdot 3 \]
\[ = 9 + 9\sqrt{3} + 6 = 15 + 9\sqrt{3} \]

### Step 4: Combine the results
Now we can combine the results from the numerator and denominator:

\[ \frac{15 + 9\sqrt{3}}{6} = \frac{15}{6} + \frac{9\sqrt{3}}{6} = \frac{5}{2} + \frac{3}{2}\sqrt{3} \]

### Step 5: Identify \(a\) and \(b\)
From the expression \(\frac{5}{2} + \frac{3}{2}\sqrt{3}\), we can identify:

\[ a = \frac{5}{2}, \quad b = \frac{3}{2} \]

### Step 6: Calculate \(a + b\)
Now we calculate \(a + b\):

\[ a + b = \frac{5}{2} + \frac{3}{2} = \frac{5 + 3}{2} = \frac{8}{2} = 4 \]

### Step 7: Calculate \(\sqrt{a + b}\)
Finally, we find \(\sqrt{a + b}\):

\[ \sqrt{a + b} = \sqrt{4} = 2 \]

### Final Answer
Thus, the value of \(\sqrt{a + b}\) is \(2\).

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