$\sqrt{18}·\sqrt{2}$
= $\sqrt{2·9}·\sqrt{2}$
= $\sqrt{9}·\sqrt{2}·\sqrt{2}$
= $3·2$
= $6$

$\frac{\sqrt{180}}{\sqrt{\mathrm{0,002}}}$
= $\sqrt{\frac{180}{0.002}}$
= $\sqrt{\frac{180.000}{2}}$
= $\sqrt{90000}$
= 300

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

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

${\left(\sqrt{5}-\sqrt{180}\right)}^2$ = ${\left(\sqrt{5}\right)}^2-2·\sqrt{5}·\sqrt{180}+{\left(\sqrt{180}\right)}^2$
= $5-2·\sqrt{5}·\sqrt{5·36}+180$
= $5-2·\sqrt{5}·\sqrt{5}·\sqrt{36}+180$
= $5-2·5·6+180$
= $5-60+180$
= $125$

$\sqrt{2x}·\sqrt{32x}$
= $\sqrt{2}·\sqrt{x}·\sqrt{32}·\sqrt{x}$
= $\sqrt{2}·\sqrt{32}·\sqrt{x}·\sqrt{x}$
= $\sqrt{2}·\sqrt{2·16}·x$
= $\sqrt{2}·\sqrt{2}·\sqrt{16}·x$
= $2·4·x$
= $8x$

Wir können die $\sqrt{14}$ wie Einheiten behandeln und einfach verrechnen.

$-4\sqrt{14}+4\sqrt{14}$
= 0