$\sqrt{180}·\sqrt{5}$
= $\sqrt{5·36}·\sqrt{5}$
= $\sqrt{36}·\sqrt{5}·\sqrt{5}$
= $6·5$
= $30$

$\sqrt{5}·\sqrt{\frac{1}{20}}$
= $\sqrt{5·\frac{1}{20}}$
= $\sqrt{1·\frac{1}{4}}$
= $\sqrt{\frac{1}{4}}$
= $\frac{1}{2}$

$\left(\sqrt{45}+\sqrt{5}\right)·\sqrt{5}$
= $\sqrt{45}·\sqrt{5}+\sqrt{5}·\sqrt{5}$
= $\sqrt{5·9}·\sqrt{5}+5$
= $\sqrt{9}·\sqrt{5}·\sqrt{5}+5$
= $3·5+5$
= $15+5$
= $20$

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

${\left(\sqrt{10}+\sqrt{360}\right)}^2$ = ${\left(\sqrt{10}\right)}^2+2·\sqrt{10}·\sqrt{360}+{\left(\sqrt{360}\right)}^2$
= $10+2·\sqrt{10}·\sqrt{10·36}+360$
= $10+2·\sqrt{10}·\sqrt{10}·\sqrt{36}+360$
= $10+2·10·6+360$
= $10+120+360$
= $490$

$\sqrt{5x}·\sqrt{80x}$
= $\sqrt{5}·\sqrt{x}·\sqrt{80}·\sqrt{x}$
= $\sqrt{5}·\sqrt{80}·\sqrt{x}·\sqrt{x}$
= $\sqrt{5}·\sqrt{5·16}·x$
= $\sqrt{5}·\sqrt{5}·\sqrt{16}·x$
= $5·4·x$
= $20x$

Wir können die $\sqrt{14}$ wie Einheiten behandeln und einfach verrechnen.

$12\sqrt{14}+6\sqrt{14}$
= $18\sqrt{14}$