$\frac{\sqrt{18}}{\sqrt{2}}$
= $\sqrt{\frac{18}{2}}$
= $\sqrt{9}$
= 3

$\sqrt{\frac{4}{3}}·\sqrt{\frac{4}{3}}$
= $\sqrt{\frac{4}{3}·\frac{4}{3}}$
= $\sqrt{\frac{16}{9}}$
= $\frac{4}{3}$

$\left(\sqrt{10}+\sqrt{250}\right)·\sqrt{10}$
= $\sqrt{10}·\sqrt{10}+\sqrt{250}·\sqrt{10}$
= $10+\sqrt{10·25}·\sqrt{10}$
= $10+\sqrt{25}·\sqrt{10}·\sqrt{10}$
= $10+5·10$
= $10+50$
= $60$

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

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

$\sqrt{81x}-2\sqrt{x}$
= $\sqrt{81}·\sqrt{x}-2\sqrt{x}$
= $9\sqrt{x}-2\sqrt{x}$
= $7\sqrt{x}$

Wir können die $\sqrt{39}$ wie Einheiten behandeln und einfach verrechnen.

$-4\sqrt{39}-\sqrt{39}$
= $-5\sqrt{39}$