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

$\frac{\sqrt{\frac{9}{8}}}{\sqrt{2}}$
= $\sqrt{\frac{\frac{9}{8}}{2}}$
= $\sqrt{\frac{9}{8}·\frac{1}{2}}$
= $\sqrt{\frac{9}{16}}$
= $\frac{3}{4}$

Wir können die $\sqrt{10}$ wie Einheiten behandeln und ausklammern, bzw. einfach verrechnen.

$-5\sqrt{10}+5\sqrt{10}$

= $\left(-5+5\right)·\sqrt{10}$

= 0

$\left(\sqrt{108}+\sqrt{3}\right)·\sqrt{3}$
= $\sqrt{108}·\sqrt{3}+\sqrt{3}·\sqrt{3}$
= $\sqrt{3·36}·\sqrt{3}+3$
= $\sqrt{36}·\sqrt{3}·\sqrt{3}+3$
= $6·3+3$
= $18+3$
= $21$

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

${\left(\sqrt{2}-\sqrt{50}\right)}^2$ = ${\left(\sqrt{2}\right)}^2-2·\sqrt{2}·\sqrt{50}+{\left(\sqrt{50}\right)}^2$
= $2-2·\sqrt{2}·\sqrt{2·25}+50$
= $2-2·\sqrt{2}·\sqrt{2}·\sqrt{25}+50$
= $2-2·2·5+50$
= $2-20+50$
= $32$

$\frac{\sqrt{360x}}{\sqrt{10x}}$
= $\frac{\sqrt{360}·\sqrt{x}}{\sqrt{10}·\sqrt{x}}$
= $\frac{\sqrt{360}}{\sqrt{10}}$
= $\sqrt{\frac{360}{10}}$
= $\sqrt{36}$
= $6$

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

${\left(\sqrt{125}-\sqrt{5}\right)}^2$ = ${\left(\sqrt{125}\right)}^2-2·\sqrt{125}·\sqrt{5}+{\left(\sqrt{5}\right)}^2$
= $125-2·\sqrt{5·25}·\sqrt{5}+5$
= $125-2·\sqrt{25}·\sqrt{5}·\sqrt{5}+5$
= $125-2·5·5+5$
= $125-50+5$
= $80$