$\frac{\sqrt{360}}{\sqrt{10}}$
= $\sqrt{\frac{360}{10}}$
= $\sqrt{36}$
= 6

$\frac{\sqrt{\frac{1}{2}}}{\sqrt{\frac{1}{2}}}$
= $\sqrt{\frac{\frac{1}{2}}{\frac{1}{2}}}$
= $\sqrt{\frac{1}{2}·2}$
= $\sqrt{1·1}$
= $\sqrt{1}$
= $1$

$\left(\sqrt{27}+\sqrt{3}\right)·\sqrt{3}$
= $\sqrt{27}·\sqrt{3}+\sqrt{3}·\sqrt{3}$
= $\sqrt{3·9}·\sqrt{3}+3$
= $\sqrt{9}·\sqrt{3}·\sqrt{3}+3$
= $3·3+3$
= $9+3$
= $12$

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

$\left(\sqrt{8}-\sqrt{3}\right)·\left(\sqrt{8}+\sqrt{3}\right)$ = ${\left(\sqrt{8}\right)}^2-{\left(\sqrt{3}\right)}^2$
= $8-3$
= $5$

$\sqrt{49x}-4\sqrt{x}$
= $\sqrt{49}·\sqrt{x}-4\sqrt{x}$
= $7\sqrt{x}-4\sqrt{x}$
= $3\sqrt{x}$

Wir können die $\sqrt{22}$ wie Einheiten behandeln und einfach verrechnen.

$-\sqrt{22}+5\sqrt{22}$
= $4\sqrt{22}$