$\frac{\sqrt{40}}{\sqrt{10}}$
= $\sqrt{\frac{40}{10}}$
= $\sqrt{4}$
= 2

$\sqrt{2}·\sqrt{\frac{1}{72}}$
= $\sqrt{2·\frac{1}{72}}$
= $\sqrt{1·\frac{1}{36}}$
= $\sqrt{\frac{1}{36}}$
= $\frac{1}{6}$

$\sqrt{5}·\left(\sqrt{180}+\sqrt{5}\right)$
= $\sqrt{5}·\sqrt{180}+\sqrt{5}·\sqrt{5}$
= $\sqrt{5}·\sqrt{5·36}+5$
= $\sqrt{5}·\sqrt{5}·\sqrt{36}+5$
= $5·6+5$
= $30+5$
= $35$

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

${\left(\sqrt{10}-\sqrt{90}\right)}^2$ = ${\left(\sqrt{10}\right)}^2-2·\sqrt{10}·\sqrt{90}+{\left(\sqrt{90}\right)}^2$
= $10-2·\sqrt{10}·\sqrt{10·9}+90$
= $10-2·\sqrt{10}·\sqrt{10}·\sqrt{9}+90$
= $10-2·10·3+90$
= $10-60+90$
= $40$

$\sqrt{50x}·\sqrt{2x}$
= $\sqrt{50}·\sqrt{x}·\sqrt{2}·\sqrt{x}$
= $\sqrt{50}·\sqrt{2}·\sqrt{x}·\sqrt{x}$
= $\sqrt{2·25}·\sqrt{2}·x$
= $\sqrt{25}·\sqrt{2}·\sqrt{2}·x$
= $5·2·x$
= $10x$

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

$-2\sqrt{42}-3\sqrt{42}$
= $-5\sqrt{42}$