$\sqrt{10}·\sqrt{160}$
= $\sqrt{10}·\sqrt{10·16}$
= $\sqrt{10}·\sqrt{10}·\sqrt{16}$
= $10·4$
= $40$

$\frac{\sqrt{5000}}{\sqrt{\mathrm{0,02}}}$
= $\sqrt{\frac{5000}{0.02}}$
= $\sqrt{\frac{500.000}{2}}$
= $\sqrt{250000}$
= 500

$\sqrt{2}·\left(\sqrt{72}+\sqrt{2}\right)$
= $\sqrt{2}·\sqrt{72}+\sqrt{2}·\sqrt{2}$
= $\sqrt{2}·\sqrt{2·36}+2$
= $\sqrt{2}·\sqrt{2}·\sqrt{36}+2$
= $2·6+2$
= $12+2$
= $14$

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

${\left(\sqrt{3}-\sqrt{108}\right)}^2$ = ${\left(\sqrt{3}\right)}^2-2·\sqrt{3}·\sqrt{108}+{\left(\sqrt{108}\right)}^2$
= $3-2·\sqrt{3}·\sqrt{3·36}+108$
= $3-2·\sqrt{3}·\sqrt{3}·\sqrt{36}+108$
= $3-2·3·6+108$
= $3-36+108$
= $75$

$-5\sqrt{x}+\sqrt{9x}$
= $-5\sqrt{x}+\sqrt{9}·\sqrt{x}$
= $-5\sqrt{x}+3\sqrt{x}$
= $-2\sqrt{x}$

Wir können die $\sqrt{13}$ wie Einheiten behandeln und einfach verrechnen.

$-5\sqrt{13}-8\sqrt{13}$
= $-13\sqrt{13}$