To simplify the expression \( \frac{3}{2 - \sqrt{3}} \) and rationalize the denominator, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2 - \sqrt{3} \) is \( 2 + \sqrt{3} \).
Here are the steps:
- Multiply numerator and denominator by the conjugate:
\[ \frac{3}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{3(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]
- Simplify the denominator using the difference of squares:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
- Now, substitute this back into the fraction:
\[ \frac{3(2 + \sqrt{3})}{1} = 3(2 + \sqrt{3}) \]
- Distribute the 3 in the numerator:
\[ 3(2 + \sqrt{3}) = 6 + 3\sqrt{3} \]
Thus, the simplified expression is:
\[ \boxed{6 + 3\sqrt{3}} \]
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