$\frac{\sqrt{160}}{\sqrt{10}}$
= $\sqrt{\frac{160}{10}}$
= $\sqrt{16}$
= 4

$\sqrt{\frac{1}{108}}·\sqrt{3}$
= $\sqrt{\frac{1}{108}·3}$
= $\sqrt{\frac{1}{36}·1}$
= $\sqrt{\frac{1}{36}}$
= $\frac{1}{6}$

$\left(\sqrt{32}+\sqrt{2}\right)·\sqrt{2}$
= $\sqrt{32}·\sqrt{2}+\sqrt{2}·\sqrt{2}$
= $\sqrt{2·16}·\sqrt{2}+2$
= $\sqrt{16}·\sqrt{2}·\sqrt{2}+2$
= $4·2+2$
= $8+2$
= $10$

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

${\left(\sqrt{5}-\sqrt{45}\right)}^2$ = ${\left(\sqrt{5}\right)}^2-2·\sqrt{5}·\sqrt{45}+{\left(\sqrt{45}\right)}^2$
= $5-2·\sqrt{5}·\sqrt{5·9}+45$
= $5-2·\sqrt{5}·\sqrt{5}·\sqrt{9}+45$
= $5-2·5·3+45$
= $5-30+45$
= $20$

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

Wir können die $\sqrt{34}$ wie Einheiten behandeln und einfach verrechnen.

$\sqrt{34}+9\sqrt{34}$
= $10\sqrt{34}$