$\frac{\sqrt{125}}{\sqrt{5}}$
= $\sqrt{\frac{125}{5}}$
= $\sqrt{25}$
= 5

$\sqrt{2}·\sqrt{\frac{1}{32}}$
= $\sqrt{2·\frac{1}{32}}$
= $\sqrt{1·\frac{1}{16}}$
= $\sqrt{\frac{1}{16}}$
= $\frac{1}{4}$

$\sqrt{3}·\left(\sqrt{3}+\sqrt{75}\right)$
= $\sqrt{3}·\sqrt{3}+\sqrt{3}·\sqrt{75}$
= $3+\sqrt{3}·\sqrt{3·25}$
= $3+\sqrt{3}·\sqrt{3}·\sqrt{25}$
= $3+3·5$
= $3+15$
= $18$

Nach der 3. binomischen Formel ((a+b)(a-b)=a²-b²) gilt:

$\left(\sqrt{3}-\sqrt{4}\right)·\left(\sqrt{3}+\sqrt{4}\right)$ = ${\left(\sqrt{3}\right)}^2-{\left(\sqrt{4}\right)}^2$
= $3-4$
= $-1$

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

Wir können die $\sqrt{24}$ wie Einheiten behandeln und einfach die $\sqrt{24}$ verrechnen.

$-10\sqrt{24}-9\sqrt{24}$
= $-19\sqrt{24}$