Express `(4+3sqrt(3))/([7+4sqrt((3))])` in the form `A+sqrt(B)`, where A and B are integers.

Answer

To express \(\frac{4 + 3\sqrt{3}}{7 + 4\sqrt{3}}\) in the form \(A + \sqrt{B}\), where \(A\) and \(B\) are integers, we will follow these steps:

### Step 1: Rationalize the Denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(7 + 4\sqrt{3}\) is \(7 - 4\sqrt{3}\).

\[ \frac{4 + 3\sqrt{3}}{7 + 4\sqrt{3}} \cdot \frac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} = \frac{(4 + 3\sqrt{3})(7 - 4\sqrt{3})}{(7 + 4\sqrt{3})(7 - 4\sqrt{3})} \]

### Step 2: Expand the Numerator
Now we will expand the numerator:

\[ (4 + 3\sqrt{3})(7 - 4\sqrt{3}) = 4 \cdot 7 + 4 \cdot (-4\sqrt{3}) + 3\sqrt{3} \cdot 7 + 3\sqrt{3} \cdot (-4\sqrt{3}) \]

Calculating each term:

- \(4 \cdot 7 = 28\)
- \(4 \cdot (-4\sqrt{3}) = -16\sqrt{3}\)
- \(3\sqrt{3} \cdot 7 = 21\sqrt{3}\)
- \(3\sqrt{3} \cdot (-4\sqrt{3}) = -12\)

Combining these, we have:

\[ 28 - 16\sqrt{3} + 21\sqrt{3} - 12 = 16 + 5\sqrt{3} \]

### Step 3: Expand the Denominator
Now we will expand the denominator:

\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1 \]

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

\[ \frac{16 + 5\sqrt{3}}{1} = 16 + 5\sqrt{3} \]

### Step 5: Write in the Form \(A + \sqrt{B}\)
We can express \(16 + 5\sqrt{3}\) in the form \(A + \sqrt{B}\):

Here, \(A = 16\) and \(B = 75\) (since \(5\sqrt{3} = \sqrt{75}\)).

Thus, the final answer is:

\[ 16 + \sqrt{75} \]

### Final Answer
The expression \(\frac{4 + 3\sqrt{3}}{7 + 4\sqrt{3}}\) in the form \(A + \sqrt{B}\) is:

\[ 16 + \sqrt{75} \]