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

$\frac{\sqrt{\mathrm{0,008}}}{\sqrt{\mathrm{0,2}}}$
= $\sqrt{\frac{0.008}{0.2}}$
= $\sqrt{\frac{0.08}{2}}$
= $\sqrt{\mathrm{0,04}}$
= 0.2

$\left(\sqrt{5}+\sqrt{180}\right)·\sqrt{5}$
= $\sqrt{5}·\sqrt{5}+\sqrt{180}·\sqrt{5}$
= $5+\sqrt{5·36}·\sqrt{5}$
= $5+\sqrt{36}·\sqrt{5}·\sqrt{5}$
= $5+6·5$
= $5+30$
= $35$

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

${\left(\sqrt{5}+\sqrt{45}\right)}^2$ = ${\left(\sqrt{5}\right)}^2+2·\sqrt{5}·\sqrt{45}+{\left(\sqrt{45}\right)}^2$
= $5+2·\sqrt{5}·\sqrt{5·9}+45$
= $5+2·\sqrt{5}·\sqrt{5}·\sqrt{9}+45$
= $5+2·5·3+45$
= $5+30+45$
= $80$

$\sqrt{3x}·\sqrt{12x}$
= $\sqrt{3}·\sqrt{x}·\sqrt{12}·\sqrt{x}$
= $\sqrt{3}·\sqrt{12}·\sqrt{x}·\sqrt{x}$
= $\sqrt{3}·\sqrt{3·4}·x$
= $\sqrt{3}·\sqrt{3}·\sqrt{4}·x$
= $3·2·x$
= $6x$

Wir können die $\sqrt{38}$ wie Einheiten behandeln und einfach verrechnen.

$8\sqrt{38}+9\sqrt{38}$
= $17\sqrt{38}$