برای ساده‌تر شدن، حاصل هر یک از عبارت‌ها را جداگانه محاسبه می‌کنیم:

${{\left( \sqrt[3]{5+{{\left( -\frac{1}{\sqrt{3}} \right)}^{4}}+{{\left( -\sqrt{2} \right)}^{2}}} \right)}^{-\frac{3}{2}}}={{\left( {{\left( 5+{{\left( -\frac{1}{\sqrt{3}} \right)}^{4}}+{{\left( -\sqrt{2} \right)}^{2}} \right)}^{\frac{1}{3}}} \right)}^{-\frac{3}{2}}}={{\left( 5+{{\left( -\frac{1}{\sqrt{3}} \right)}^{4}}+{{\left( -\sqrt{2} \right)}^{2}} \right)}^{-\frac{1}{2}}}$ 

$={{\left( 5+\frac{1}{9}+2 \right)}^{-\frac{1}{2}}}={{\left( \frac{64}{9} \right)}^{-\frac{1}{2}}}=\sqrt{\frac{9}{64}}=\frac{3}{8}$ 

${{\left( \sqrt{2\frac{1}{4}} \right)}^{3}}={{\left( \sqrt{\frac{9}{4}} \right)}^{3}}={{\left( \frac{3}{2} \right)}^{3}}=\frac{27}{8}\Rightarrow {{\left( \sqrt[3]{5+{{\left( -\frac{1}{\sqrt{3}} \right)}^{4}}+{{\left( -\sqrt{2} \right)}^{2}}} \right)}^{-\frac{3}{2}}}+{{\left( \sqrt{2\frac{1}{4}} \right)}^{3}}=\frac{3}{8}+\frac{27}{8}=\frac{30}{8}=3\frac{6}{8}=3/75$