$\frac{\sqrt{125}}{\sqrt{5}}$
= $\sqrt{\frac{125}{5}}$
= $\sqrt{25}$
= 5

$\sqrt{\mathrm{0,2}}·\sqrt{\mathrm{0,008}}$
= $\sqrt{\mathrm{0,2}·\mathrm{0,008}}$
= $\sqrt{\mathrm{0,0016}}$
= 0.04

$\left(\sqrt{180}+\sqrt{5}\right)·\sqrt{5}$
= $\sqrt{180}·\sqrt{5}+\sqrt{5}·\sqrt{5}$
= $\sqrt{5·36}·\sqrt{5}+5$
= $\sqrt{36}·\sqrt{5}·\sqrt{5}+5$
= $6·5+5$
= $30+5$
= $35$

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

${\left(\sqrt{10}+\sqrt{90}\right)}^2$ = ${\left(\sqrt{10}\right)}^2+2·\sqrt{10}·\sqrt{90}+{\left(\sqrt{90}\right)}^2$
= $10+2·\sqrt{10}·\sqrt{10·9}+90$
= $10+2·\sqrt{10}·\sqrt{10}·\sqrt{9}+90$
= $10+2·10·3+90$
= $10+60+90$
= $160$

$\frac{\sqrt{8x}}{\sqrt{2x}}$
= $\frac{\sqrt{8}·\sqrt{x}}{\sqrt{2}·\sqrt{x}}$
= $\frac{\sqrt{8}}{\sqrt{2}}$
= $\sqrt{\frac{8}{2}}$
= $\sqrt{4}$
= $2$

Wir können die $\sqrt{40}$ wie Einheiten behandeln und einfach verrechnen.

$5\sqrt{40}+\sqrt{40}$
= $6\sqrt{40}$