$\sqrt{3}·\sqrt{48}$
= $\sqrt{3}·\sqrt{3·16}$
= $\sqrt{3}·\sqrt{3}·\sqrt{16}$
= $3·4$
= $12$

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

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

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

${\left(\sqrt{5}-\sqrt{45}\right)}^2$ = ${\left(\sqrt{5}\right)}^2-2·\sqrt{5}·\sqrt{45}+{\left(\sqrt{45}\right)}^2$
= $5-2·\sqrt{5}·\sqrt{5·9}+45$
= $5-2·\sqrt{5}·\sqrt{5}·\sqrt{9}+45$
= $5-2·5·3+45$
= $5-30+45$
= $20$

$\sqrt{108x}·\sqrt{3x}$
= $\sqrt{108}·\sqrt{x}·\sqrt{3}·\sqrt{x}$
= $\sqrt{108}·\sqrt{3}·\sqrt{x}·\sqrt{x}$
= $\sqrt{3·36}·\sqrt{3}·x$
= $\sqrt{36}·\sqrt{3}·\sqrt{3}·x$
= $6·3·x$
= $18x$

Wir können die $\sqrt{32}$ wie Einheiten behandeln und einfach verrechnen.

$-4\sqrt{32}+6\sqrt{32}$
= $2\sqrt{32}$