The value of ` ( 4 + 3 sqrt(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: Multiply by the Conjugate
We will multiply both the numerator and the denominator by the conjugate of the denominator, which 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: Simplify the Denominator
Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):

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

### Step 3: 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:

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

Now combine these results:

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

### Step 4: Combine Like Terms
Combine the constant terms and the terms with \(\sqrt{3}\):

- Constant terms: \( 28 - 12 = 16 \)
- Terms with \(\sqrt{3}\): \( -16\sqrt{3} + 21\sqrt{3} = 5\sqrt{3} \)

So the numerator simplifies to:

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

### Step 5: Final Result
Putting it all together, we have:

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

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

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