$a){\left(\sqrt{7}+1\right)}^2={\sqrt{7}}^2+2\cdot \sqrt{7}\cdot 1+1^2=7+2\sqrt{7}+1=8+2\sqrt{7}$

$b){\left(\sqrt{5}-3\right)}^2={\sqrt{5}}^2-2\cdot \sqrt{5}\cdot 3+3^2=5-6\sqrt{5}+9=14-6\sqrt{5}$

$c){\left(6-\sqrt{3}\right)}^2=6^2-2\cdot 6\cdot \sqrt{3}+{\sqrt{3}}^2=36-12\sqrt{3}+3=39-12\sqrt{3}$

$d){\left(1+2\sqrt{2}\right)}^2=1^2+2\cdot 1\cdot 2\sqrt{2}+{\left(2\sqrt{2}\right)}^2=1+4\sqrt{2}+4\cdot 2=9+4\sqrt{2}$

$e){\left(\sqrt{3}+\sqrt{2}\right)}^2={\sqrt{3}}^2+2\cdot \sqrt{3}\cdot \sqrt{2}+{\sqrt{2}}^2=3+2\sqrt{6}+2=5+2\sqrt{6}$

$f){\left(\sqrt{3}-2\sqrt{2}\right)}^2={\sqrt{3}}^2-2\cdot \sqrt{3}\cdot 2\sqrt{2}+{\left(2\sqrt{2}\right)}^2=3-4\sqrt{6}+4\cdot 2=11-4\sqrt{6}$

$g){\left(\sqrt{6}+\sqrt{15}\right)}^2={\sqrt{6}}^2+2\cdot \sqrt{6}\cdot \sqrt{15}+{\sqrt{15}}^2=6+2\sqrt{90}+15=21+2\cdot \sqrt{9}\cdot \sqrt{10}=21+2\cdot 3\sqrt{10}=21+6\sqrt{10}$

$h){\left(\frac{\sqrt{2}}{2}-\sqrt{6}\right)}^2={\left(\frac{\sqrt{2}}{2}\right)}^2-2\cdot \frac{\sqrt{2}}{2}\cdot \sqrt{6}+{\sqrt{6}}^2=\frac{2}{4}-\sqrt{12}+6=\frac{1}{2}-\sqrt{4}\cdot \sqrt{3}+6=6\frac{1}{2}-2\sqrt{3}$