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

$\frac{\sqrt{\frac{3}{10}}}{\sqrt{\frac{8}{15}}}$
= $\sqrt{\frac{\frac{3}{10}}{\frac{8}{15}}}$
= $\sqrt{\frac{3}{10}·\frac{15}{8}}$
= $\sqrt{\frac{3}{2}·\frac{3}{8}}$
= $\sqrt{\frac{9}{16}}$
= $\frac{3}{4}$

$\sqrt{3}·\left(\sqrt{27}+\sqrt{3}\right)$
= $\sqrt{3}·\sqrt{27}+\sqrt{3}·\sqrt{3}$
= $\sqrt{3}·\sqrt{3·9}+3$
= $\sqrt{3}·\sqrt{3}·\sqrt{9}+3$
= $3·3+3$
= $9+3$
= $12$

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

${\left(\sqrt{5}+\sqrt{80}\right)}^2$ = ${\left(\sqrt{5}\right)}^2+2·\sqrt{5}·\sqrt{80}+{\left(\sqrt{80}\right)}^2$
= $5+2·\sqrt{5}·\sqrt{5·16}+80$
= $5+2·\sqrt{5}·\sqrt{5}·\sqrt{16}+80$
= $5+2·5·4+80$
= $5+40+80$
= $125$

$\frac{\sqrt{40x}}{\sqrt{10x}}$
= $\frac{\sqrt{40}·\sqrt{x}}{\sqrt{10}·\sqrt{x}}$
= $\frac{\sqrt{40}}{\sqrt{10}}$
= $\sqrt{\frac{40}{10}}$
= $\sqrt{4}$
= $2$

Wir können die $\sqrt{40}$ wie Einheiten behandeln und einfach die $\sqrt{40}$ verrechnen.

$5\sqrt{40}+8\sqrt{40}$
= $13\sqrt{40}$