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

$\frac{\sqrt{\frac{1}{6}}}{\sqrt{\frac{3}{2}}}$
= $\sqrt{\frac{\frac{1}{6}}{\frac{3}{2}}}$
= $\sqrt{\frac{1}{6}·\frac{2}{3}}$
= $\sqrt{\frac{1}{3}·\frac{1}{3}}$
= $\sqrt{\frac{1}{9}}$
= $\frac{1}{3}$

$\sqrt{10}·\left(\sqrt{360}+\sqrt{10}\right)$
= $\sqrt{10}·\sqrt{360}+\sqrt{10}·\sqrt{10}$
= $\sqrt{10}·\sqrt{10·36}+10$
= $\sqrt{10}·\sqrt{10}·\sqrt{36}+10$
= $10·6+10$
= $60+10$
= $70$

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

${\left(\sqrt{2}-\sqrt{32}\right)}^2$ = ${\left(\sqrt{2}\right)}^2-2·\sqrt{2}·\sqrt{32}+{\left(\sqrt{32}\right)}^2$
= $2-2·\sqrt{2}·\sqrt{2·16}+32$
= $2-2·\sqrt{2}·\sqrt{2}·\sqrt{16}+32$
= $2-2·2·4+32$
= $2-16+32$
= $18$

$\frac{\sqrt{48x}}{\sqrt{3x}}$
= $\frac{\sqrt{48}·\sqrt{x}}{\sqrt{3}·\sqrt{x}}$
= $\frac{\sqrt{48}}{\sqrt{3}}$
= $\sqrt{\frac{48}{3}}$
= $\sqrt{16}$
= $4$

Wir können die $\sqrt{42}$ wie Einheiten behandeln und einfach verrechnen.

$15\sqrt{42}+8\sqrt{42}$
= $23\sqrt{42}$