Let's find which choice is equivalent to the given fraction by rationalizing the denominator and simplifying the expression.
Given:
[tex]\[
\frac{\sqrt{12}}{\sqrt{3} + 3}
\][/tex]
The first step is to rationalize the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{3} - 3\)[/tex]:
[tex]\[
\frac{\sqrt{12}(\sqrt{3} - 3)}{(\sqrt{3} + 3)(\sqrt{3} - 3)}
\][/tex]
Next, let's simplify the denominator:
[tex]\[
(\sqrt{3} + 3)(\sqrt{3} - 3) = (\sqrt{3})^2 - 3^2 = 3 - 9 = -6
\][/tex]
Now, we simplify the numerator:
[tex]\[
\sqrt{12} (\sqrt{3} - 3) = \sqrt{12} \cdot \sqrt{3} - \sqrt{12} \cdot 3
\][/tex]
Note that:
[tex]\[
\sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3}
\][/tex]
Therefore:
[tex]\[
\sqrt{12} \cdot \sqrt{3} = 2 \sqrt{3} \cdot \sqrt{3} = 2 \cdot 3 = 6
\][/tex]
and
[tex]\[
\sqrt{12} \cdot 3 = 2 \sqrt{3} \cdot 3 = 6 \sqrt{3}
\][/tex]
So, the numerator becomes:
[tex]\[
6 - 6\sqrt{3}
\][/tex]
Putting it all together, we get:
[tex]\[
\frac{6 - 6\sqrt{3}}{-6}
\][/tex]
We can simplify this expression further by dividing both terms in the numerator by -6:
[tex]\[
\frac{6}{-6} - \frac{6\sqrt{3}}{-6} = -1 + \sqrt{3}
\][/tex]
Thus, the simplified expression is:
[tex]\[
-1 + \sqrt{3}
\][/tex]
Therefore, the correct choice equivalent to the given fraction is:
[tex]\[
\boxed{-1+\sqrt{3}}
\][/tex]
So, the answer is:
D. [tex]\(-1+\sqrt{3}\)[/tex]
