$\frac{\sqrt{48}}{\sqrt{3}}$
= $\sqrt{\frac{48}{3}}$
= $\sqrt{16}$
= 4

$\sqrt{\frac{250}{9}}·\sqrt{\frac{1}{10}}$
= $\sqrt{\frac{250}{9}·\frac{1}{10}}$
= $\sqrt{\frac{25}{9}·1}$
= $\sqrt{\frac{25}{9}}$
= $\frac{5}{3}$

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

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

$\left(\sqrt{9}+\sqrt{3}\right)·\left(\sqrt{9}-\sqrt{3}\right)$ = ${\left(\sqrt{9}\right)}^2-{\left(\sqrt{3}\right)}^2$
= $9-3$
= $6$

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

Wir können die $\sqrt{45}$ wie Einheiten behandeln und einfach verrechnen.

$8\sqrt{45}-2\sqrt{45}$
= $6\sqrt{45}$