If `(4+3sqrt(3))/(sqrt(7+4sqrt(3)))=A+sqrt(B)` then B-A is

Options

A. `-13`
B. `2sqrt(13)`
C. `13`
D. `3sqrt(3)-sqrt(7)`

Answer

To solve the equation \(\frac{4 + 3\sqrt{3}}{\sqrt{7 + 4\sqrt{3}}} = A + \sqrt{B}\) and find \(B - A\), we can follow these steps:

### Step 1: Simplify the denominator
We start with the denominator \(\sqrt{7 + 4\sqrt{3}}\). We can express \(7 + 4\sqrt{3}\) in a different form. Notice that:
\[ 7 + 4\sqrt{3} = (2 + \sqrt{3})^2 \]
This is because:
\[ (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]
Thus, we have:
\[ \sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3} \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression:
\[ \frac{4 + 3\sqrt{3}}{2 + \sqrt{3}} \]

### Step 3: Rationalize the denominator
To eliminate the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(2 - \sqrt{3}\):
\[ \frac{(4 + 3\sqrt{3})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} \]
The denominator simplifies as follows:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
So, the denominator becomes 1.

### Step 4: Simplify the numerator
Now we simplify the numerator:
\[ (4 + 3\sqrt{3})(2 - \sqrt{3}) = 4 \cdot 2 - 4\sqrt{3} + 3\sqrt{3} \cdot 2 - 3\sqrt{3} \cdot \sqrt{3} \]
Calculating this gives:
\[ = 8 - 4\sqrt{3} + 6\sqrt{3} - 3 = 8 - 3 + (6\sqrt{3} - 4\sqrt{3}) = 5 + 2\sqrt{3} \]

### Step 5: Final expression
Thus, we have:
\[ \frac{4 + 3\sqrt{3}}{\sqrt{7 + 4\sqrt{3}}} = 5 + 2\sqrt{3} \]
This means \(A = 5\) and \(B = 12\) (since \(2\sqrt{3} = \sqrt{12}\)).

### Step 6: Calculate \(B - A\)
Now we calculate \(B - A\):
\[ B - A = 12 - 5 = 7 \]

### Final Answer
Thus, the value of \(B - A\) is \(7\).

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