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

$\sqrt{\frac{1}{2}}·\sqrt{72}$
= $\sqrt{\frac{1}{2}·72}$
= $\sqrt{1·36}$
= $\sqrt{36}$
= $6$

$\left(\sqrt{2}+\sqrt{8}\right)·\sqrt{2}$
= $\sqrt{2}·\sqrt{2}+\sqrt{8}·\sqrt{2}$
= $2+\sqrt{2·4}·\sqrt{2}$
= $2+\sqrt{4}·\sqrt{2}·\sqrt{2}$
= $2+2·2$
= $2+4$
= $6$

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

${\left(\sqrt{10}+\sqrt{250}\right)}^2$ = ${\left(\sqrt{10}\right)}^2+2·\sqrt{10}·\sqrt{250}+{\left(\sqrt{250}\right)}^2$
= $10+2·\sqrt{10}·\sqrt{10·25}+250$
= $10+2·\sqrt{10}·\sqrt{10}·\sqrt{25}+250$
= $10+2·10·5+250$
= $10+100+250$
= $360$

$\sqrt{125x}·\sqrt{5x}$
= $\sqrt{125}·\sqrt{x}·\sqrt{5}·\sqrt{x}$
= $\sqrt{125}·\sqrt{5}·\sqrt{x}·\sqrt{x}$
= $\sqrt{5·25}·\sqrt{5}·x$
= $\sqrt{25}·\sqrt{5}·\sqrt{5}·x$
= $5·5·x$
= $25x$

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

${\left(\sqrt{32}+\sqrt{2}\right)}^2$ = ${\left(\sqrt{32}\right)}^2+2·\sqrt{32}·\sqrt{2}+{\left(\sqrt{2}\right)}^2$
= $32+2·\sqrt{2·16}·\sqrt{2}+2$
= $32+2·\sqrt{16}·\sqrt{2}·\sqrt{2}+2$
= $32+2·4·2+2$
= $32+16+2$
= $50$