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

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

$\sqrt{3}·\left(\sqrt{27}+\sqrt{3}\right)$
= $\sqrt{3}·\sqrt{27}+\sqrt{3}·\sqrt{3}$
= $\sqrt{3}·\sqrt{3·9}+3$
= $\sqrt{3}·\sqrt{3}·\sqrt{9}+3$
= $3·3+3$
= $9+3$
= $12$

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

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

$\sqrt{5x}·\sqrt{45x}$
= $\sqrt{5}·\sqrt{x}·\sqrt{45}·\sqrt{x}$
= $\sqrt{5}·\sqrt{45}·\sqrt{x}·\sqrt{x}$
= $\sqrt{5}·\sqrt{5·9}·x$
= $\sqrt{5}·\sqrt{5}·\sqrt{9}·x$
= $5·3·x$
= $15x$

Wir können die $\sqrt{37}$ wie Einheiten behandeln und einfach verrechnen.

$2\sqrt{37}+9\sqrt{37}$
= $11\sqrt{37}$