$\sqrt{10}·\sqrt{90}$
= $\sqrt{10}·\sqrt{10·9}$
= $\sqrt{10}·\sqrt{10}·\sqrt{9}$
= $10·3$
= $30$

$\sqrt{\mathrm{0,3}}·\sqrt{\mathrm{0,027}}$
= $\sqrt{\mathrm{0,3}·\mathrm{0,027}}$
= $\sqrt{\mathrm{0,0081}}$
= 0.09

$\left(\sqrt{2}+\sqrt{18}\right)·\sqrt{2}$
= $\sqrt{2}·\sqrt{2}+\sqrt{18}·\sqrt{2}$
= $2+\sqrt{2·9}·\sqrt{2}$
= $2+\sqrt{9}·\sqrt{2}·\sqrt{2}$
= $2+3·2$
= $2+6$
= $8$

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

${\left(\sqrt{10}-\sqrt{40}\right)}^2$ = ${\left(\sqrt{10}\right)}^2-2·\sqrt{10}·\sqrt{40}+{\left(\sqrt{40}\right)}^2$
= $10-2·\sqrt{10}·\sqrt{10·4}+40$
= $10-2·\sqrt{10}·\sqrt{10}·\sqrt{4}+40$
= $10-2·10·2+40$
= $10-40+40$
= $10$

$\frac{\sqrt{250x}}{\sqrt{10x}}$
= $\frac{\sqrt{250}·\sqrt{x}}{\sqrt{10}·\sqrt{x}}$
= $\frac{\sqrt{250}}{\sqrt{10}}$
= $\sqrt{\frac{250}{10}}$
= $\sqrt{25}$
= $5$

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

$-3\sqrt{45}+\sqrt{45}$
= $-2\sqrt{45}$