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

$\frac{\sqrt{\frac{1}{36}}}{\sqrt{1}}$
= $\sqrt{\frac{\frac{1}{36}}{1}}$
= $\sqrt{\frac{1}{36}·1}$
= $\sqrt{\frac{1}{36}}$
= $\frac{1}{6}$

$\left(\sqrt{12}+\sqrt{3}\right)·\sqrt{3}$
= $\sqrt{12}·\sqrt{3}+\sqrt{3}·\sqrt{3}$
= $\sqrt{3·4}·\sqrt{3}+3$
= $\sqrt{4}·\sqrt{3}·\sqrt{3}+3$
= $2·3+3$
= $6+3$
= $9$

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

$\left(\sqrt{6}+\sqrt{9}\right)·\left(\sqrt{6}-\sqrt{9}\right)$ = ${\left(\sqrt{6}\right)}^2-{\left(\sqrt{9}\right)}^2$
= $6-9$
= $-3$

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

Wir können die $\sqrt{30}$ wie Einheiten behandeln und einfach verrechnen.

$-14\sqrt{30}-4\sqrt{30}$
= $-18\sqrt{30}$