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

$\sqrt{2}·\sqrt{\frac{1}{72}}$
= $\sqrt{2·\frac{1}{72}}$
= $\sqrt{1·\frac{1}{36}}$
= $\sqrt{\frac{1}{36}}$
= $\frac{1}{6}$

$\left(\sqrt{5}+\sqrt{125}\right)·\sqrt{5}$
= $\sqrt{5}·\sqrt{5}+\sqrt{125}·\sqrt{5}$
= $5+\sqrt{5·25}·\sqrt{5}$
= $5+\sqrt{25}·\sqrt{5}·\sqrt{5}$
= $5+5·5$
= $5+25$
= $30$

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

${\left(\sqrt{5}-\sqrt{80}\right)}^2$ = ${\left(\sqrt{5}\right)}^2-2·\sqrt{5}·\sqrt{80}+{\left(\sqrt{80}\right)}^2$
= $5-2·\sqrt{5}·\sqrt{5·16}+80$
= $5-2·\sqrt{5}·\sqrt{5}·\sqrt{16}+80$
= $5-2·5·4+80$
= $5-40+80$
= $45$

$\sqrt{2x}·\sqrt{8x}$
= $\sqrt{2}·\sqrt{x}·\sqrt{8}·\sqrt{x}$
= $\sqrt{2}·\sqrt{8}·\sqrt{x}·\sqrt{x}$
= $\sqrt{2}·\sqrt{2·4}·x$
= $\sqrt{2}·\sqrt{2}·\sqrt{4}·x$
= $2·2·x$
= $4x$

Wir können die $\sqrt{50}$ wie Einheiten behandeln und einfach verrechnen.

$\sqrt{50}+4\sqrt{50}$
= $5\sqrt{50}$