$\frac{\sqrt{180}}{\sqrt{5}}$
= $\sqrt{\frac{180}{5}}$
= $\sqrt{36}$
= 6

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

$\sqrt{10}·\left(\sqrt{10}+\sqrt{160}\right)$
= $\sqrt{10}·\sqrt{10}+\sqrt{10}·\sqrt{160}$
= $10+\sqrt{10}·\sqrt{10·16}$
= $10+\sqrt{10}·\sqrt{10}·\sqrt{16}$
= $10+10·4$
= $10+40$
= $50$

Nach der 1. binomischen Formel ((a+b)²=a²+2ab+b²) gilt:

${\left(\sqrt{3}+\sqrt{27}\right)}^2$ = ${\left(\sqrt{3}\right)}^2+2·\sqrt{3}·\sqrt{27}+{\left(\sqrt{27}\right)}^2$
= $3+2·\sqrt{3}·\sqrt{3·9}+27$
= $3+2·\sqrt{3}·\sqrt{3}·\sqrt{9}+27$
= $3+2·3·3+27$
= $3+18+27$
= $48$

$\sqrt{81x}+4\sqrt{x}$
= $\sqrt{81}·\sqrt{x}+4\sqrt{x}$
= $9\sqrt{x}+4\sqrt{x}$
= $13\sqrt{x}$

Wir können die $\sqrt{43}$ wie Einheiten behandeln und einfach verrechnen.

$-7\sqrt{43}-\sqrt{43}$
= $-8\sqrt{43}$