Step 1

Multiplying the numerator and the denominator with the conjugate,

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}\times{\frac{{{2}+\sqrt{{{3}}}}}{{{2}+\sqrt{{{3}}}}}}\)

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={\frac{{{\left({3}\right)}{\left({2}+\sqrt{{{3}}}\right)}}}{{{\left({2}-\sqrt{{{3}}}\right)}{\left({2}+\sqrt{{{3}}}\right)}}}}\)

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={\frac{{{6}+{3}\sqrt{{{3}}}}}{{{\left({2}\right)}^{{{2}}}-{\left(\sqrt{{{3}}}\right)}^{{{2}}}}}}\ {i}.{e}.\ {\left({a}+{b}\right)}{\left({a}-{b}\right)}={a}^{{{2}}}-{b}^{{{2}}}\)

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={\frac{{{6}+{3}\sqrt{{{3}}}}}{{{4}-{3}}}}\) (Since square and square root are opposite operations, they cancel out)

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={6}+{3}\sqrt{{{3}}}\)

Multiplying the numerator and the denominator with the conjugate,

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}\times{\frac{{{2}+\sqrt{{{3}}}}}{{{2}+\sqrt{{{3}}}}}}\)

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={\frac{{{\left({3}\right)}{\left({2}+\sqrt{{{3}}}\right)}}}{{{\left({2}-\sqrt{{{3}}}\right)}{\left({2}+\sqrt{{{3}}}\right)}}}}\)

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={\frac{{{6}+{3}\sqrt{{{3}}}}}{{{\left({2}\right)}^{{{2}}}-{\left(\sqrt{{{3}}}\right)}^{{{2}}}}}}\ {i}.{e}.\ {\left({a}+{b}\right)}{\left({a}-{b}\right)}={a}^{{{2}}}-{b}^{{{2}}}\)

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={\frac{{{6}+{3}\sqrt{{{3}}}}}{{{4}-{3}}}}\) (Since square and square root are opposite operations, they cancel out)

\(\displaystyle{\frac{{{3}}}{{{2}-\sqrt{{{3}}}}}}={6}+{3}\sqrt{{{3}}}\)