$\frac{\sqrt{50}}{\sqrt{2}}$
= $\sqrt{\frac{50}{2}}$
= $\sqrt{25}$
= 5

$\sqrt{2}·\sqrt{\frac{1}{32}}$
= $\sqrt{2·\frac{1}{32}}$
= $\sqrt{1·\frac{1}{16}}$
= $\sqrt{\frac{1}{16}}$
= $\frac{1}{4}$

$\sqrt{10}·\left(\sqrt{10}+\sqrt{160}\right)$
= $\sqrt{10}·\sqrt{10}+\sqrt{10}·\sqrt{160}$
= $10+\sqrt{10}·\sqrt{10·16}$
= $10+\sqrt{10}·\sqrt{10}·\sqrt{16}$
= $10+10·4$
= $10+40$
= $50$

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

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

$\frac{\sqrt{90x}}{\sqrt{10x}}$
= $\frac{\sqrt{90}·\sqrt{x}}{\sqrt{10}·\sqrt{x}}$
= $\frac{\sqrt{90}}{\sqrt{10}}$
= $\sqrt{\frac{90}{10}}$
= $\sqrt{9}$
= $3$

Wir können die $\sqrt{45}$ wie Einheiten behandeln und einfach verrechnen.

$-5\sqrt{45}-3\sqrt{45}$
= $-8\sqrt{45}$