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

$\sqrt{\frac{50}{3}}·\sqrt{\frac{1}{6}}$
= $\sqrt{\frac{50}{3}·\frac{1}{6}}$
= $\sqrt{\frac{25}{3}·\frac{1}{3}}$
= $\sqrt{\frac{25}{9}}$
= $\frac{5}{3}$

$\left(\sqrt{45}+\sqrt{5}\right)·\sqrt{5}$
= $\sqrt{45}·\sqrt{5}+\sqrt{5}·\sqrt{5}$
= $\sqrt{5·9}·\sqrt{5}+5$
= $\sqrt{9}·\sqrt{5}·\sqrt{5}+5$
= $3·5+5$
= $15+5$
= $20$

Nach der 1. binomischen Formel ((a+b)²=a²+2ab+b²) gilt:

${\left(\sqrt{3}+\sqrt{108}\right)}^2$ = ${\left(\sqrt{3}\right)}^2+2·\sqrt{3}·\sqrt{108}+{\left(\sqrt{108}\right)}^2$
= $3+2·\sqrt{3}·\sqrt{3·36}+108$
= $3+2·\sqrt{3}·\sqrt{3}·\sqrt{36}+108$
= $3+2·3·6+108$
= $3+36+108$
= $147$

$\sqrt{75x}·\sqrt{3x}$
= $\sqrt{75}·\sqrt{x}·\sqrt{3}·\sqrt{x}$
= $\sqrt{75}·\sqrt{3}·\sqrt{x}·\sqrt{x}$
= $\sqrt{3·25}·\sqrt{3}·x$
= $\sqrt{25}·\sqrt{3}·\sqrt{3}·x$
= $5·3·x$
= $15x$

Wir können die $\sqrt{7}$ wie Einheiten behandeln und einfach verrechnen.

$5\sqrt{7}+7\sqrt{7}$
= $12\sqrt{7}$