$\sqrt{5}·\sqrt{180}$
= $\sqrt{5}·\sqrt{5·36}$
= $\sqrt{5}·\sqrt{5}·\sqrt{36}$
= $5·6$
= $30$

$\sqrt{5}·\sqrt{\frac{1}{45}}$
= $\sqrt{5·\frac{1}{45}}$
= $\sqrt{1·\frac{1}{9}}$
= $\sqrt{\frac{1}{9}}$
= $\frac{1}{3}$

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

Nach der 3. binomischen Formel ((a+b)(a-b)=a²-b²) gilt:

$\left(\sqrt{3}+\sqrt{5}\right)·\left(\sqrt{3}-\sqrt{5}\right)$ = ${\left(\sqrt{3}\right)}^2-{\left(\sqrt{5}\right)}^2$
= $3-5$
= $-2$

$\sqrt{8x}·\sqrt{2x}$
= $\sqrt{8}·\sqrt{x}·\sqrt{2}·\sqrt{x}$
= $\sqrt{8}·\sqrt{2}·\sqrt{x}·\sqrt{x}$
= $\sqrt{2·4}·\sqrt{2}·x$
= $\sqrt{4}·\sqrt{2}·\sqrt{2}·x$
= $2·2·x$
= $4x$

Wir können die $\sqrt{12}$ wie Einheiten behandeln und einfach verrechnen.

$13\sqrt{12}+6\sqrt{12}$
= $19\sqrt{12}$