$\frac{\sqrt{20}}{\sqrt{5}}$
= $\sqrt{\frac{20}{5}}$
= $\sqrt{4}$
= 2

$\sqrt{\frac{1}{18}}·\sqrt{2}$
= $\sqrt{\frac{1}{18}·2}$
= $\sqrt{\frac{1}{9}·1}$
= $\sqrt{\frac{1}{9}}$
= $\frac{1}{3}$

$\left(\sqrt{160}+\sqrt{10}\right)·\sqrt{10}$
= $\sqrt{160}·\sqrt{10}+\sqrt{10}·\sqrt{10}$
= $\sqrt{10·16}·\sqrt{10}+10$
= $\sqrt{16}·\sqrt{10}·\sqrt{10}+10$
= $4·10+10$
= $40+10$
= $50$

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

${\left(\sqrt{3}-\sqrt{108}\right)}^2$ = ${\left(\sqrt{3}\right)}^2-2·\sqrt{3}·\sqrt{108}+{\left(\sqrt{108}\right)}^2$
= $3-2·\sqrt{3}·\sqrt{3·36}+108$
= $3-2·\sqrt{3}·\sqrt{3}·\sqrt{36}+108$
= $3-2·3·6+108$
= $3-36+108$
= $75$

$\sqrt{5x}·\sqrt{20x}$
= $\sqrt{5}·\sqrt{x}·\sqrt{20}·\sqrt{x}$
= $\sqrt{5}·\sqrt{20}·\sqrt{x}·\sqrt{x}$
= $\sqrt{5}·\sqrt{5·4}·x$
= $\sqrt{5}·\sqrt{5}·\sqrt{4}·x$
= $5·2·x$
= $10x$

Wir können die $\sqrt{12}$ wie Einheiten behandeln und einfach verrechnen.

$2\sqrt{12}-5\sqrt{12}$
= $-3\sqrt{12}$