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

$\frac{\sqrt{\frac{3}{20}}}{\sqrt{\frac{1}{15}}}$
= $\sqrt{\frac{\frac{3}{20}}{\frac{1}{15}}}$
= $\sqrt{\frac{3}{20}·15}$
= $\sqrt{\frac{3}{4}·3}$
= $\sqrt{\frac{9}{4}}$
= $\frac{3}{2}$

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

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

$\left(\sqrt{9}+5\right)·\left(\sqrt{9}-5\right)$ = ${\left(\sqrt{9}\right)}^2-5^2$
= $9-25$
= $-16$

$\sqrt{4x}-5\sqrt{x}$
= $\sqrt{4}·\sqrt{x}-5\sqrt{x}$
= $2\sqrt{x}-5\sqrt{x}$
= $-3\sqrt{x}$

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

${\left(\sqrt{180}-\sqrt{5}\right)}^2$ = ${\left(\sqrt{180}\right)}^2-2·\sqrt{180}·\sqrt{5}+{\left(\sqrt{5}\right)}^2$
= $180-2·\sqrt{5·36}·\sqrt{5}+5$
= $180-2·\sqrt{36}·\sqrt{5}·\sqrt{5}+5$
= $180-2·6·5+5$
= $180-60+5$
= $125$