$\frac{\sqrt{360}}{\sqrt{10}}$
= $\sqrt{\frac{360}{10}}$
= $\sqrt{36}$
= 6

$\sqrt{\frac{45}{2}}·\sqrt{\frac{1}{10}}$
= $\sqrt{\frac{45}{2}·\frac{1}{10}}$
= $\sqrt{\frac{9}{2}·\frac{1}{2}}$
= $\sqrt{\frac{9}{4}}$
= $\frac{3}{2}$

$\sqrt{2}·\left(\sqrt{8}+\sqrt{2}\right)$
= $\sqrt{2}·\sqrt{8}+\sqrt{2}·\sqrt{2}$
= $\sqrt{2}·\sqrt{2·4}+2$
= $\sqrt{2}·\sqrt{2}·\sqrt{4}+2$
= $2·2+2$
= $4+2$
= $6$

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

${\left(\sqrt{2}-\sqrt{18}\right)}^2$ = ${\left(\sqrt{2}\right)}^2-2·\sqrt{2}·\sqrt{18}+{\left(\sqrt{18}\right)}^2$
= $2-2·\sqrt{2}·\sqrt{2·9}+18$
= $2-2·\sqrt{2}·\sqrt{2}·\sqrt{9}+18$
= $2-2·2·3+18$
= $2-12+18$
= $8$

$4\sqrt{x}+\sqrt{49x}$
= $4\sqrt{x}+\sqrt{49}·\sqrt{x}$
= $4\sqrt{x}+7\sqrt{x}$
= $11\sqrt{x}$

Wir können die $\sqrt{39}$ wie Einheiten behandeln und einfach die $\sqrt{39}$ verrechnen.

$-10\sqrt{39}-8\sqrt{39}$
= $-18\sqrt{39}$