$\frac{\sqrt{12}}{\sqrt{3}}$
= $\sqrt{\frac{12}{3}}$
= $\sqrt{4}$
= 2

$\sqrt{\mathrm{4,8}}·\sqrt{\mathrm{0,003}}$
= $\sqrt{\mathrm{4,8}·\mathrm{0,003}}$
= $\sqrt{\mathrm{0,0144}}$
= 0.12

$\sqrt{3}·\left(\sqrt{3}+\sqrt{108}\right)$
= $\sqrt{3}·\sqrt{3}+\sqrt{3}·\sqrt{108}$
= $3+\sqrt{3}·\sqrt{3·36}$
= $3+\sqrt{3}·\sqrt{3}·\sqrt{36}$
= $3+3·6$
= $3+18$
= $21$

Nach der 1. binomischen Formel ((a+b)²=a²+2ab+b²) gilt:

${\left(\sqrt{3}+\sqrt{108}\right)}^2$ = ${\left(\sqrt{3}\right)}^2+2·\sqrt{3}·\sqrt{108}+{\left(\sqrt{108}\right)}^2$
= $3+2·\sqrt{3}·\sqrt{3·36}+108$
= $3+2·\sqrt{3}·\sqrt{3}·\sqrt{36}+108$
= $3+2·3·6+108$
= $3+36+108$
= $147$

$\sqrt{3x}·\sqrt{27x}$
= $\sqrt{3}·\sqrt{x}·\sqrt{27}·\sqrt{x}$
= $\sqrt{3}·\sqrt{27}·\sqrt{x}·\sqrt{x}$
= $\sqrt{3}·\sqrt{3·9}·x$
= $\sqrt{3}·\sqrt{3}·\sqrt{9}·x$
= $3·3·x$
= $9x$

Wir können die $\sqrt{5}$ wie Einheiten behandeln und einfach verrechnen.

$6\sqrt{5}+2\sqrt{5}$
= $8\sqrt{5}$