$\sqrt{10}·\sqrt{40}$
= $\sqrt{10}·\sqrt{10·4}$
= $\sqrt{10}·\sqrt{10}·\sqrt{4}$
= $10·2$
= $20$

$\frac{\sqrt{\frac{1}{30}}}{\sqrt{\frac{3}{10}}}$
= $\sqrt{\frac{\frac{1}{30}}{\frac{3}{10}}}$
= $\sqrt{\frac{1}{30}·\frac{10}{3}}$
= $\sqrt{\frac{1}{3}·\frac{1}{3}}$
= $\sqrt{\frac{1}{9}}$
= $\frac{1}{3}$

$\left(\sqrt{20}+\sqrt{5}\right)·\sqrt{5}$
= $\sqrt{20}·\sqrt{5}+\sqrt{5}·\sqrt{5}$
= $\sqrt{5·4}·\sqrt{5}+5$
= $\sqrt{4}·\sqrt{5}·\sqrt{5}+5$
= $2·5+5$
= $10+5$
= $15$

Nach der 3. binomischen Formel ((a+b)(a-b)=a²-b²) gilt:

$\left(\sqrt{2}-\sqrt{6}\right)·\left(\sqrt{2}+\sqrt{6}\right)$ = ${\left(\sqrt{2}\right)}^2-{\left(\sqrt{6}\right)}^2$
= $2-6$
= $-4$

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

Wir können die $\sqrt{27}$ wie Einheiten behandeln und einfach verrechnen.

$-3\sqrt{27}-8\sqrt{27}$
= $-11\sqrt{27}$