Es ist

${\displaystyle {}{\begin{aligned}X^{2}&={\left({\sqrt {3}}+{\sqrt {7}}\right)}^{2}\\&={\sqrt {3}}^{2}+{\sqrt {7}}^{2}+2{\sqrt {3}}\cdot {\sqrt {7}}\\&=3+7+2{\sqrt {21}}\\&=10+2{\sqrt {21}}\end{aligned}}}$

und

${\displaystyle {}{\begin{aligned}X^{4}&={\left(X^{2}\right)}^{2}\\&={\left(10+2{\sqrt {21}}\right)}^{2}\\&=100+4\cdot 21+40{\sqrt {21}}\\&=184+40{\sqrt {21}}.\end{aligned}}}$

Somit ist

${\displaystyle {}{\begin{aligned}X^{4}-20X^{2}+16&=184+40{\sqrt {21}}-20{\left(10+2{\sqrt {21}}\right)}+16\\&=184-200+16+{\left(40-20\cdot 2\right)}{\sqrt {21}}\\&=0.\end{aligned}}}$