$\frac{\sqrt{160}}{\sqrt{10}}$
= $\sqrt{\frac{160}{10}}$
= $\sqrt{16}$
= 4

$\sqrt{\frac{8}{5}}·\sqrt{\frac{5}{2}}$
= $\sqrt{\frac{8}{5}·\frac{5}{2}}$
= $\sqrt{4·1}$
= $\sqrt{4}$
= $2$

$\sqrt{2}·\left(\sqrt{72}+\sqrt{2}\right)$
= $\sqrt{2}·\sqrt{72}+\sqrt{2}·\sqrt{2}$
= $\sqrt{2}·\sqrt{2·36}+2$
= $\sqrt{2}·\sqrt{2}·\sqrt{36}+2$
= $2·6+2$
= $12+2$
= $14$

Nach der 1. binomischen Formel ((a+b)²=a²+2ab+b²) gilt:

${\left(\sqrt{3}+\sqrt{75}\right)}^2$ = ${\left(\sqrt{3}\right)}^2+2·\sqrt{3}·\sqrt{75}+{\left(\sqrt{75}\right)}^2$
= $3+2·\sqrt{3}·\sqrt{3·25}+75$
= $3+2·\sqrt{3}·\sqrt{3}·\sqrt{25}+75$
= $3+2·3·5+75$
= $3+30+75$
= $108$

$3\sqrt{x}+\sqrt{81x}$
= $3\sqrt{x}+\sqrt{81}·\sqrt{x}$
= $3\sqrt{x}+9\sqrt{x}$
= $12\sqrt{x}$

Nach der 1. binomischen Formel ((a+b)²=a²+2ab+b²) gilt:

${\left(\sqrt{45}+\sqrt{5}\right)}^2$ = ${\left(\sqrt{45}\right)}^2+2·\sqrt{45}·\sqrt{5}+{\left(\sqrt{5}\right)}^2$
= $45+2·\sqrt{5·9}·\sqrt{5}+5$
= $45+2·\sqrt{9}·\sqrt{5}·\sqrt{5}+5$
= $45+2·3·5+5$
= $45+30+5$
= $80$