\( \frac{1}{2-\sqrt{3}}+\frac{1}{3-2\sqrt{2}}+\frac{1}{4-\sqrt{15}}+\frac{1}{5-2\sqrt{6}} \)
multiply the numerator and denominator of each fraction by its conjunction, so we have:
\( \frac{2+\sqrt{3}}{4-3}+\frac{3+2\sqrt{2}}{9-8}+\frac{4+\sqrt{15}}{16-15}+\frac{5+2\sqrt{6}}{25-24}=2+3+4+5+2\sqrt{2}+\sqrt{3}+2\sqrt{6}+\sqrt{15}=14+2\sqrt{2}+\sqrt{3}+2\sqrt{6}+\sqrt{15} \)
\( \Rightarrow a=14 , b=2 , c=1 , d=2 \Rightarrow \frac{\sqrt{a+d}}{bc}=\frac{\sqrt{14+2}}{2 \times 1}=\frac{\sqrt{16}}{2}=\frac{4}{2}=2 \)
