To simplify and rationalize the expression \(-\frac{\sqrt{3}}{8 + \sqrt{3}}\), we will first multiply the numerator and the denominator by the conjugate of the denominator.
The conjugate of \(8 + \sqrt{3}\) is \(8 - \sqrt{3}\).
Now, let's perform the multiplication:
\[ -\frac{\sqrt{3}}{8 + \sqrt{3}} \cdot \frac{8 - \sqrt{3}}{8 - \sqrt{3}} = -\frac{\sqrt{3} (8 - \sqrt{3})}{(8 + \sqrt{3})(8 - \sqrt{3})} \]
Now, calculate the denominator using the difference of squares:
\[ (8 + \sqrt{3})(8 - \sqrt{3}) = 8^2 - (\sqrt{3})^2 = 64 - 3 = 61 \]
Next, calculate the numerator:
\[ -\sqrt{3} (8 - \sqrt{3}) = -8\sqrt{3} + 3 \]
Putting it all together, we have:
\[ -\frac{8\sqrt{3} - 3}{61} = \frac{3 - 8\sqrt{3}}{61} \]
So the simplified and rationalized expression is:
\[ \frac{3 - 8\sqrt{3}}{61} \]