Wzór na taki pierwiastek pod pierwiastkiem:
\sqrt{a\pm b \sqrt{c}}=\frac{\sqrt{a+x}}{\sqrt2}\pm\frac{\sqrt{a-x}}{\sqrt2}
gdzie
x=\sqrt{a^2-(b \sqrt{c})^2}
{(11-\sqrt{21})^{\frac{1}{2}}+ (11+\sqrt{21})^{\frac{1}{2}}=\sqrt{11-\sqrt{21}}+\sqrt{11+\sqrt{21}}=}
{=\frac{\sqrt{11+\sqrt{11-(\sqrt{21})^2}}}{\sqrt2}-\frac{\sqrt{11-\sqrt11^2-(\sqrt{21})^2}}{\sqrt2}+\frac{\sqrt{11+\sqrt{11^2-(\sqrt{21})^2}}}{\sqrt2}+\frac{\sqrt{11-\sqrt{11^2-(\sqrt{21})^2}}}{\sqrt2}=}
{=\frac{\sqrt{11+\sqrt{121-21}}}{\sqrt2}-\frac{\sqrt{11-\sqrt{121-21}}}{\sqrt2}+\frac{\sqrt{11-\sqrt{121-21}}}{\sqrt2}+\frac{\sqrt{11-\sqrt{121-21}}}{\sqrt2}=}
{=\frac{\sqrt{11+\sqrt{100}}}{\sqrt2}-\frac{\sqrt{11-\sqrt{100}}}{\sqrt2}+\frac{\sqrt{11+\sqrt{100}}}{\sqrt2}+\frac{\sqrt{11-\sqrt{100}}}{\sqrt2}=}
{=\frac{\sqrt{11+10}-\sqrt{11-10}+\sqrt{11+10}+\sqrt{11-10}}{\sqrt2}=\frac{\sqrt{21}-\sqrt{1}+\sqrt{21}+\sqrt1}{\sqrt2}=\frac{2\sqrt{21}}{\sqrt2}=\frac{2\sqrt{21}*\sqrt2}{\sqrt2*\sqrt2}=}
=\frac{2\sqrt{42}}{2}=\sqrt{42}