$\frac{\sqrt{108}}{\sqrt{3}}$
= $\sqrt{\frac{108}{3}}$
= $\sqrt{36}$
= 6

$\sqrt{\frac{1}{27}}·\sqrt{3}$
= $\sqrt{\frac{1}{27}·3}$
= $\sqrt{\frac{1}{9}·1}$
= $\sqrt{\frac{1}{9}}$
= $\frac{1}{3}$

$\left(\sqrt{72}+\sqrt{2}\right)·\sqrt{2}$
= $\sqrt{72}·\sqrt{2}+\sqrt{2}·\sqrt{2}$
= $\sqrt{2·36}·\sqrt{2}+2$
= $\sqrt{36}·\sqrt{2}·\sqrt{2}+2$
= $6·2+2$
= $12+2$
= $14$

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

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

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

Wir können die $\sqrt{13}$ wie Einheiten behandeln und einfach die $\sqrt{13}$ verrechnen.

$16\sqrt{13}+9\sqrt{13}$
= $25\sqrt{13}$