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

$\frac{\sqrt{\frac{2}{5}}}{\sqrt{\frac{5}{2}}}$
= $\sqrt{\frac{\frac{2}{5}}{\frac{5}{2}}}$
= $\sqrt{\frac{2}{5}·\frac{2}{5}}$
= $\sqrt{\frac{4}{25}}$
= $\frac{2}{5}$

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

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

${\left(\sqrt{2}-\sqrt{18}\right)}^2$ = ${\left(\sqrt{2}\right)}^2-2·\sqrt{2}·\sqrt{18}+{\left(\sqrt{18}\right)}^2$
= $2-2·\sqrt{2}·\sqrt{2·9}+18$
= $2-2·\sqrt{2}·\sqrt{2}·\sqrt{9}+18$
= $2-2·2·3+18$
= $2-12+18$
= $8$

$\frac{\sqrt{180x}}{\sqrt{5x}}$
= $\frac{\sqrt{180}·\sqrt{x}}{\sqrt{5}·\sqrt{x}}$
= $\frac{\sqrt{180}}{\sqrt{5}}$
= $\sqrt{\frac{180}{5}}$
= $\sqrt{36}$
= $6$

Wir können die $\sqrt{12}$ wie Einheiten behandeln und einfach verrechnen.

$15\sqrt{12}+9\sqrt{12}$
= $24\sqrt{12}$