Question

Simplify ${(\sqrt[6]{621}-\sqrt{6\frac{3}{4}})}^2$

Answer 1

${(\sqrt[6]{621}-\sqrt{6\frac{3}{4}})}^2={({\left(21\right)}^{\frac{1}{6}}-\sqrt{\frac{27}{4}})}^2={({\left(21\right)}^{\frac{1}{6}}-\frac{3\sqrt{3}}{2})}^2\because {(a-b)}^2=a^2-2ab+b^{2}a={\left(21\right)}^{\frac{1}{6}},b=\frac{3\sqrt{3}}{2}={\left[{\left(21\right)}^{1/6}\right]}^2-2{\left(21\right)}^{1/6}\frac{3\sqrt{3}}{2}+{\left[\frac{3\sqrt{3}}{2}\right]}^2={\left(21\right)}^{1/3}-3{\left(21\right)}^{1/6}\sqrt{3}+\frac{27}{4}=\sqrt[3]{321}-3\sqrt{3}\sqrt[6]{621}+\frac{27}{4}$