$\sqrt{10}·\sqrt{90}$
= $\sqrt{10}·\sqrt{10·9}$
= $\sqrt{10}·\sqrt{10}·\sqrt{9}$
= $10·3$
= $30$

$\sqrt{3}·\sqrt{\frac{3}{4}}$
= $\sqrt{3·\frac{3}{4}}$
= $\sqrt{\frac{9}{4}}$
= $\frac{3}{2}$

$\sqrt{2}·\left(\sqrt{32}+\sqrt{2}\right)$
= $\sqrt{2}·\sqrt{32}+\sqrt{2}·\sqrt{2}$
= $\sqrt{2}·\sqrt{2·16}+2$
= $\sqrt{2}·\sqrt{2}·\sqrt{16}+2$
= $2·4+2$
= $8+2$
= $10$

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

$\left(\sqrt{3}-8\right)·\left(\sqrt{3}+8\right)$ = ${\left(\sqrt{3}\right)}^2-8^2$
= $3-64$
= $-61$

$\frac{\sqrt{108x}}{\sqrt{3x}}$
= $\frac{\sqrt{108}·\sqrt{x}}{\sqrt{3}·\sqrt{x}}$
= $\frac{\sqrt{108}}{\sqrt{3}}$
= $\sqrt{\frac{108}{3}}$
= $\sqrt{36}$
= $6$

Wir können die $\sqrt{13}$ wie Einheiten behandeln und einfach verrechnen.

$-5\sqrt{13}+3\sqrt{13}$
= $-2\sqrt{13}$