$\sqrt{10}·\sqrt{360}$
= $\sqrt{10}·\sqrt{10·36}$
= $\sqrt{10}·\sqrt{10}·\sqrt{36}$
= $10·6$
= $60$

$\sqrt{\frac{1}{32}}·\sqrt{2}$
= $\sqrt{\frac{1}{32}·2}$
= $\sqrt{\frac{1}{16}·1}$
= $\sqrt{\frac{1}{16}}$
= $\frac{1}{4}$

$\left(\sqrt{80}+\sqrt{5}\right)·\sqrt{5}$
= $\sqrt{80}·\sqrt{5}+\sqrt{5}·\sqrt{5}$
= $\sqrt{5·16}·\sqrt{5}+5$
= $\sqrt{16}·\sqrt{5}·\sqrt{5}+5$
= $4·5+5$
= $20+5$
= $25$

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

${\left(\sqrt{2}+\sqrt{8}\right)}^2$ = ${\left(\sqrt{2}\right)}^2+2·\sqrt{2}·\sqrt{8}+{\left(\sqrt{8}\right)}^2$
= $2+2·\sqrt{2}·\sqrt{2·4}+8$
= $2+2·\sqrt{2}·\sqrt{2}·\sqrt{4}+8$
= $2+2·2·2+8$
= $2+8+8$
= $18$

$\sqrt{10x}·\sqrt{250x}$
= $\sqrt{10}·\sqrt{x}·\sqrt{250}·\sqrt{x}$
= $\sqrt{10}·\sqrt{250}·\sqrt{x}·\sqrt{x}$
= $\sqrt{10}·\sqrt{10·25}·x$
= $\sqrt{10}·\sqrt{10}·\sqrt{25}·x$
= $10·5·x$
= $50x$

Wir können die $\sqrt{22}$ wie Einheiten behandeln und einfach die $\sqrt{22}$ verrechnen.

$-\sqrt{22}-2\sqrt{22}$
= $-3\sqrt{22}$