$\sqrt{27}·\sqrt{3}$
= $\sqrt{3·9}·\sqrt{3}$
= $\sqrt{9}·\sqrt{3}·\sqrt{3}$
= $3·3$
= $9$

$\frac{\sqrt{\frac{3}{4}}}{\sqrt{\frac{4}{3}}}$
= $\sqrt{\frac{\frac{3}{4}}{\frac{4}{3}}}$
= $\sqrt{\frac{3}{4}·\frac{3}{4}}$
= $\sqrt{\frac{9}{16}}$
= $\frac{3}{4}$

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

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

$\left(\sqrt{2}+3\right)·\left(\sqrt{2}-3\right)$ = ${\left(\sqrt{2}\right)}^2-3^2$
= $2-9$
= $-7$

$\frac{\sqrt{160x}}{\sqrt{10x}}$
= $\frac{\sqrt{160}·\sqrt{x}}{\sqrt{10}·\sqrt{x}}$
= $\frac{\sqrt{160}}{\sqrt{10}}$
= $\sqrt{\frac{160}{10}}$
= $\sqrt{16}$
= $4$

Wir können die $\sqrt{8}$ wie Einheiten behandeln und einfach verrechnen.

$3\sqrt{8}+7\sqrt{8}$
= $10\sqrt{8}$