$\sqrt{3}·\sqrt{108}$
= $\sqrt{3}·\sqrt{3·36}$
= $\sqrt{3}·\sqrt{3}·\sqrt{36}$
= $3·6$
= $18$

$\sqrt{\mathrm{0,003}}·\sqrt{\mathrm{4,8}}$
= $\sqrt{\mathrm{0,003}·\mathrm{4,8}}$
= $\sqrt{\mathrm{0,0144}}$
= 0.12

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

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

$\left(\sqrt{9}+8\right)·\left(\sqrt{9}-8\right)$ = ${\left(\sqrt{9}\right)}^2-8^2$
= $9-64$
= $-55$

$\sqrt{3x}·\sqrt{12x}$
= $\sqrt{3}·\sqrt{x}·\sqrt{12}·\sqrt{x}$
= $\sqrt{3}·\sqrt{12}·\sqrt{x}·\sqrt{x}$
= $\sqrt{3}·\sqrt{3·4}·x$
= $\sqrt{3}·\sqrt{3}·\sqrt{4}·x$
= $3·2·x$
= $6x$

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

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