$\frac{\sqrt{80}}{\sqrt{5}}$
= $\sqrt{\frac{80}{5}}$
= $\sqrt{16}$
= 4

$\sqrt{8}·\sqrt{2}$
= $\sqrt{8·2}$
= $\sqrt{16}$
= $4$

$\left(\sqrt{5}+\sqrt{20}\right)·\sqrt{5}$
= $\sqrt{5}·\sqrt{5}+\sqrt{20}·\sqrt{5}$
= $5+\sqrt{5·4}·\sqrt{5}$
= $5+\sqrt{4}·\sqrt{5}·\sqrt{5}$
= $5+2·5$
= $5+10$
= $15$

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

${\left(\sqrt{10}-\sqrt{250}\right)}^2$ = ${\left(\sqrt{10}\right)}^2-2·\sqrt{10}·\sqrt{250}+{\left(\sqrt{250}\right)}^2$
= $10-2·\sqrt{10}·\sqrt{10·25}+250$
= $10-2·\sqrt{10}·\sqrt{10}·\sqrt{25}+250$
= $10-2·10·5+250$
= $10-100+250$
= $160$

$\sqrt{81x}-5\sqrt{x}$
= $\sqrt{81}·\sqrt{x}-5\sqrt{x}$
= $9\sqrt{x}-5\sqrt{x}$
= $4\sqrt{x}$

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

$16\sqrt{42}+7\sqrt{42}$
= $23\sqrt{42}$