$\sqrt{2}·\sqrt{32}$
= $\sqrt{2}·\sqrt{2·16}$
= $\sqrt{2}·\sqrt{2}·\sqrt{16}$
= $2·4$
= $8$

$\sqrt{\frac{6}{5}}·\sqrt{\frac{10}{3}}$
= $\sqrt{\frac{6}{5}·\frac{10}{3}}$
= $\sqrt{2·2}$
= $\sqrt{4}$
= $2$

$\sqrt{10}·\left(\sqrt{250}+\sqrt{10}\right)$
= $\sqrt{10}·\sqrt{250}+\sqrt{10}·\sqrt{10}$
= $\sqrt{10}·\sqrt{10·25}+10$
= $\sqrt{10}·\sqrt{10}·\sqrt{25}+10$
= $10·5+10$
= $50+10$
= $60$

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

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

$-2\sqrt{x}+\sqrt{64x}$
= $-2\sqrt{x}+\sqrt{64}·\sqrt{x}$
= $-2\sqrt{x}+8\sqrt{x}$
= $6\sqrt{x}$

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

${\left(\sqrt{20}-\sqrt{5}\right)}^2$ = ${\left(\sqrt{20}\right)}^2-2·\sqrt{20}·\sqrt{5}+{\left(\sqrt{5}\right)}^2$
= $20-2·\sqrt{5·4}·\sqrt{5}+5$
= $20-2·\sqrt{4}·\sqrt{5}·\sqrt{5}+5$
= $20-2·2·5+5$
= $20-20+5$
= $5$