$\frac{\sqrt{27}}{\sqrt{3}}$
= $\sqrt{\frac{27}{3}}$
= $\sqrt{9}$
= 3

$\sqrt{5}·\sqrt{\frac{1}{180}}$
= $\sqrt{5·\frac{1}{180}}$
= $\sqrt{1·\frac{1}{36}}$
= $\sqrt{\frac{1}{36}}$
= $\frac{1}{6}$

$\left(\sqrt{90}+\sqrt{10}\right)·\sqrt{10}$
= $\sqrt{90}·\sqrt{10}+\sqrt{10}·\sqrt{10}$
= $\sqrt{10·9}·\sqrt{10}+10$
= $\sqrt{9}·\sqrt{10}·\sqrt{10}+10$
= $3·10+10$
= $30+10$
= $40$

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

${\left(\sqrt{2}+\sqrt{72}\right)}^2$ = ${\left(\sqrt{2}\right)}^2+2·\sqrt{2}·\sqrt{72}+{\left(\sqrt{72}\right)}^2$
= $2+2·\sqrt{2}·\sqrt{2·36}+72$
= $2+2·\sqrt{2}·\sqrt{2}·\sqrt{36}+72$
= $2+2·2·6+72$
= $2+24+72$
= $98$

$\sqrt{3x}·\sqrt{108x}$
= $\sqrt{3}·\sqrt{x}·\sqrt{108}·\sqrt{x}$
= $\sqrt{3}·\sqrt{108}·\sqrt{x}·\sqrt{x}$
= $\sqrt{3}·\sqrt{3·36}·x$
= $\sqrt{3}·\sqrt{3}·\sqrt{36}·x$
= $3·6·x$
= $18x$

Wir können die $\sqrt{15}$ wie Einheiten behandeln und einfach verrechnen.

$3\sqrt{15}+5\sqrt{15}$
= $8\sqrt{15}$