$\frac{\sqrt{48}}{\sqrt{3}}$
= $\sqrt{\frac{48}{3}}$
= $\sqrt{16}$
= 4

$\sqrt{5}·\sqrt{5}$
= $\sqrt{5·5}$
= $\sqrt{25}$
= $5$

$\sqrt{10}·\left(\sqrt{90}+\sqrt{10}\right)$
= $\sqrt{10}·\sqrt{90}+\sqrt{10}·\sqrt{10}$
= $\sqrt{10}·\sqrt{10·9}+10$
= $\sqrt{10}·\sqrt{10}·\sqrt{9}+10$
= $10·3+10$
= $30+10$
= $40$

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

${\left(\sqrt{5}-\sqrt{45}\right)}^2$ = ${\left(\sqrt{5}\right)}^2-2·\sqrt{5}·\sqrt{45}+{\left(\sqrt{45}\right)}^2$
= $5-2·\sqrt{5}·\sqrt{5·9}+45$
= $5-2·\sqrt{5}·\sqrt{5}·\sqrt{9}+45$
= $5-2·5·3+45$
= $5-30+45$
= $20$

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

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

${\left(\sqrt{18}+\sqrt{2}\right)}^2$ = ${\left(\sqrt{18}\right)}^2+2·\sqrt{18}·\sqrt{2}+{\left(\sqrt{2}\right)}^2$
= $18+2·\sqrt{2·9}·\sqrt{2}+2$
= $18+2·\sqrt{9}·\sqrt{2}·\sqrt{2}+2$
= $18+2·3·2+2$
= $18+12+2$
= $32$