Simplify : `sqrt(((6+2sqrt(3))/(33-19sqrt(3))))`

Options

A. `5-2sqrt(3)`
B. `5+2sqrt(3)`
C. `5-3sqrt(3)`
D. `5+3sqrt(3)`

Answer

To simplify the expression \( \sqrt{\frac{6 + 2\sqrt{3}}{33 - 19\sqrt{3}}} \), we will follow these steps:

### Step 1: Rationalize the Denominator
We start by rationalizing the denominator. We can multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 33 + 19\sqrt{3} \).

\[ \sqrt{\frac{(6 + 2\sqrt{3})(33 + 19\sqrt{3})}{(33 - 19\sqrt{3})(33 + 19\sqrt{3})}} \]

### Step 2: Simplify the Denominator
Next, we simplify the denominator using the difference of squares formula:

\[ (33 - 19\sqrt{3})(33 + 19\sqrt{3}) = 33^2 - (19\sqrt{3})^2 = 1089 - 1083 = 6 \]

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

\[ (6 + 2\sqrt{3})(33 + 19\sqrt{3}) = 6 \cdot 33 + 6 \cdot 19\sqrt{3} + 2\sqrt{3} \cdot 33 + 2\sqrt{3} \cdot 19\sqrt{3} \]

Calculating each term:

- \( 6 \cdot 33 = 198 \)
- \( 6 \cdot 19\sqrt{3} = 114\sqrt{3} \)
- \( 2\sqrt{3} \cdot 33 = 66\sqrt{3} \)
- \( 2\sqrt{3} \cdot 19\sqrt{3} = 38 \cdot 3 = 114 \)

Adding these together:

\[ 198 + (114\sqrt{3} + 66\sqrt{3}) + 114 = 198 + 180\sqrt{3} + 114 = 312 + 180\sqrt{3} \]

### Step 4: Combine the Results
Now we can combine the results:

\[ \sqrt{\frac{312 + 180\sqrt{3}}{6}} = \sqrt{52 + 30\sqrt{3}} \]

### Step 5: Express as a Perfect Square
Next, we will express \( 52 + 30\sqrt{3} \) as a perfect square. We can assume it is of the form \( (a + b\sqrt{3})^2 \).

Expanding \( (a + b\sqrt{3})^2 \):

\[ a^2 + 2ab\sqrt{3} + 3b^2 \]

We need to match coefficients:

1. \( a^2 + 3b^2 = 52 \)
2. \( 2ab = 30 \)

From the second equation, we can express \( ab \):

\[ ab = 15 \implies b = \frac{15}{a} \]

Substituting \( b \) in the first equation:

\[ a^2 + 3\left(\frac{15}{a}\right)^2 = 52 \]

This simplifies to:

\[ a^2 + \frac{675}{a^2} = 52 \]

Multiplying through by \( a^2 \):

\[ a^4 - 52a^2 + 675 = 0 \]

Let \( x = a^2 \):

\[ x^2 - 52x + 675 = 0 \]

Using the quadratic formula:

\[ x = \frac{52 \pm \sqrt{52^2 - 4 \cdot 675}}{2} \]

Calculating the discriminant:

\[ 52^2 - 2700 = 2704 - 2700 = 4 \]

Thus:

\[ x = \frac{52 \pm 2}{2} = 26 \pm 1 \]

So, \( x = 27 \) or \( x = 25 \). Therefore, \( a^2 = 27 \) or \( a^2 = 25 \).

Taking square roots:

- If \( a = 5 \), then \( b = 3 \).
- If \( a = 3\sqrt{3} \), then \( b = 5/\sqrt{3} \) (not applicable here).

Thus, we have:

\[ \sqrt{52 + 30\sqrt{3}} = 5 + 3\sqrt{3} \]

### Final Answer
Finally, we take the square root:

\[ \sqrt{52 + 30\sqrt{3}} = 5 + 3\sqrt{3} \]

So, the simplified expression is:

\[ \boxed{5 + 3\sqrt{3}} \]