The value of `(4+3sqrt(3))/(7+4 sqrt(3))` is

Options

A. `5 sqrt(3)-8`
B. `5 sqrt(3) +8`
C. `8 sqrt(3) +5`
D. `8 sqrt(3) -5`

Answer

To find the value of \(\frac{4 + 3\sqrt{3}}{7 + 4\sqrt{3}}\), we will rationalize the denominator. Here are the steps:

### Step 1: Rationalize the Denominator
To rationalize the denominator, we multiply both 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:

\[ (4 + 3\sqrt{3})(7 - 4\sqrt{3}) = 4 \cdot 7 + 4 \cdot (-4\sqrt{3}) + 3\sqrt{3} \cdot 7 + 3\sqrt{3} \cdot (-4\sqrt{3}) \]
\[ = 28 - 16\sqrt{3} + 21\sqrt{3} - 12 \]
\[ = 28 - 12 + (21\sqrt{3} - 16\sqrt{3}) = 16 + 5\sqrt{3} \]

### Step 3: Expand the Denominator
Next, we will expand the denominator using the difference of squares formula:

\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 \]
\[ = 49 - 16 \cdot 3 = 49 - 48 = 1 \]

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

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

Thus, the value of \(\frac{4 + 3\sqrt{3}}{7 + 4\sqrt{3}}\) is:

\[ \boxed{16 + 5\sqrt{3}} \]