$\frac{\sqrt{50}}{\sqrt{2}}$
= $\sqrt{\frac{50}{2}}$
= $\sqrt{25}$
= 5

$\sqrt{\frac{1}{75}}·\sqrt{3}$
= $\sqrt{\frac{1}{75}·3}$
= $\sqrt{\frac{1}{25}·1}$
= $\sqrt{\frac{1}{25}}$
= $\frac{1}{5}$

$\sqrt{10}·\left(\sqrt{360}+\sqrt{10}\right)$
= $\sqrt{10}·\sqrt{360}+\sqrt{10}·\sqrt{10}$
= $\sqrt{10}·\sqrt{10·36}+10$
= $\sqrt{10}·\sqrt{10}·\sqrt{36}+10$
= $10·6+10$
= $60+10$
= $70$

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

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

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

Wir können die $\sqrt{19}$ wie Einheiten behandeln und einfach verrechnen.

$-\sqrt{19}+5\sqrt{19}$
= $4\sqrt{19}$