$\frac{\sqrt{90}}{\sqrt{10}}$
= $\sqrt{\frac{90}{10}}$
= $\sqrt{9}$
= 3

$\sqrt{10}·\sqrt{\frac{1}{250}}$
= $\sqrt{10·\frac{1}{250}}$
= $\sqrt{1·\frac{1}{25}}$
= $\sqrt{\frac{1}{25}}$
= $\frac{1}{5}$

$\sqrt{10}·\left(\sqrt{10}+\sqrt{40}\right)$
= $\sqrt{10}·\sqrt{10}+\sqrt{10}·\sqrt{40}$
= $10+\sqrt{10}·\sqrt{10·4}$
= $10+\sqrt{10}·\sqrt{10}·\sqrt{4}$
= $10+10·2$
= $10+20$
= $30$

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

$\left(\sqrt{6}-5\right)·\left(\sqrt{6}+5\right)$ = ${\left(\sqrt{6}\right)}^2-5^2$
= $6-25$
= $-19$

$2\sqrt{x}+\sqrt{81x}$
= $2\sqrt{x}+\sqrt{81}·\sqrt{x}$
= $2\sqrt{x}+9\sqrt{x}$
= $11\sqrt{x}$

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

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