$\sqrt{10}·\sqrt{40}$
= $\sqrt{10}·\sqrt{10·4}$
= $\sqrt{10}·\sqrt{10}·\sqrt{4}$
= $10·2$
= $20$

$\sqrt{\frac{6}{5}}·\sqrt{\frac{10}{3}}$
= $\sqrt{\frac{6}{5}·\frac{10}{3}}$
= $\sqrt{2·2}$
= $\sqrt{4}$
= $2$

$\left(\sqrt{10}+\sqrt{360}\right)·\sqrt{10}$
= $\sqrt{10}·\sqrt{10}+\sqrt{360}·\sqrt{10}$
= $10+\sqrt{10·36}·\sqrt{10}$
= $10+\sqrt{36}·\sqrt{10}·\sqrt{10}$
= $10+6·10$
= $10+60$
= $70$

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

${\left(\sqrt{2}+\sqrt{72}\right)}^2$ = ${\left(\sqrt{2}\right)}^2+2·\sqrt{2}·\sqrt{72}+{\left(\sqrt{72}\right)}^2$
= $2+2·\sqrt{2}·\sqrt{2·36}+72$
= $2+2·\sqrt{2}·\sqrt{2}·\sqrt{36}+72$
= $2+2·2·6+72$
= $2+24+72$
= $98$

$\sqrt{72x}·\sqrt{2x}$
= $\sqrt{72}·\sqrt{x}·\sqrt{2}·\sqrt{x}$
= $\sqrt{72}·\sqrt{2}·\sqrt{x}·\sqrt{x}$
= $\sqrt{2·36}·\sqrt{2}·x$
= $\sqrt{36}·\sqrt{2}·\sqrt{2}·x$
= $6·2·x$
= $12x$

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

$2\sqrt{45}+4\sqrt{45}$
= $6\sqrt{45}$