$\sqrt{3}·\sqrt{108}$
= $\sqrt{3}·\sqrt{3·36}$
= $\sqrt{3}·\sqrt{3}·\sqrt{36}$
= $3·6$
= $18$

$\frac{\sqrt{8}}{\sqrt{\mathrm{0,5}}}$
= $\sqrt{\frac{8}{0.5}}$
= $\sqrt{\frac{80}{5}}$
= $\sqrt{16}$
= 4

Wir können die $\sqrt{34}$ wie Einheiten behandeln und ausklammern, bzw. einfach verrechnen.

$-10\sqrt{34}-4\sqrt{34}$

= $\left(-10-4\right)·\sqrt{34}$

= $-14\sqrt{34}$

$\left(\sqrt{40}+\sqrt{10}\right)·\sqrt{10}$
= $\sqrt{40}·\sqrt{10}+\sqrt{10}·\sqrt{10}$
= $\sqrt{10·4}·\sqrt{10}+10$
= $\sqrt{4}·\sqrt{10}·\sqrt{10}+10$
= $2·10+10$
= $20+10$
= $30$

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

${\left(\sqrt{3}-\sqrt{75}\right)}^2$ = ${\left(\sqrt{3}\right)}^2-2·\sqrt{3}·\sqrt{75}+{\left(\sqrt{75}\right)}^2$
= $3-2·\sqrt{3}·\sqrt{3·25}+75$
= $3-2·\sqrt{3}·\sqrt{3}·\sqrt{25}+75$
= $3-2·3·5+75$
= $3-30+75$
= $48$

$\sqrt{49x}+4\sqrt{x}$
= $\sqrt{49}·\sqrt{x}+4\sqrt{x}$
= $7\sqrt{x}+4\sqrt{x}$
= $11\sqrt{x}$

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

${\left(\sqrt{80}-\sqrt{5}\right)}^2$ = ${\left(\sqrt{80}\right)}^2-2·\sqrt{80}·\sqrt{5}+{\left(\sqrt{5}\right)}^2$
= $80-2·\sqrt{5·16}·\sqrt{5}+5$
= $80-2·\sqrt{16}·\sqrt{5}·\sqrt{5}+5$
= $80-2·4·5+5$
= $80-40+5$
= $45$