Rút gọn các biểu thức sau:

$$\begin{gathered} a) \sqrt{\frac{2a}{3}} .\sqrt{\frac{3a}{8}}với a\ge 0; \\ b) \sqrt{13a} .\sqrt{\frac{52}{a}}với a>0; \\ c) \sqrt{5a}.\sqrt{45a}-3a với a\ge 0; \\ d) {\left(3-a\right)}^2-\sqrt{0,2}.\sqrt{180a^2} \end{gathered}$$

Hướng dẫn giải:

$\mathrm{a}) \sqrt{\frac{2\mathrm{a}}{3}} .\sqrt{\frac{3\mathrm{a}}{8}} = \sqrt{\frac{2\mathrm{a}}{3}.\frac{3\mathrm{a}}{8}} = \sqrt{\frac{{\mathrm{a}}^2}{4}} = \sqrt{{\left(\frac{\mathrm{a}}{2}\right)}^2} = \left|\frac{\mathrm{a}}{2}\right| (\mathrm{vì} \mathrm{a}\ge 0).$

$\mathrm{b}) \sqrt{13\mathrm{a}} .\sqrt{\frac{52}{\mathrm{a}}} = \sqrt{13\mathrm{a}.\frac{52}{\mathrm{a}}} = \sqrt{13.13.4} = \sqrt{{26}^2} = 26 (\mathrm{vì} \mathrm{a}>0).$

$\mathrm{c}) \sqrt{5\mathrm{a}}.\sqrt{45\mathrm{a}}-3\mathrm{a} = \sqrt{15\mathrm{a}.45\mathrm{a}}-3\mathrm{a} = \sqrt{{\left(15\mathrm{a}\right)}^2}-3\mathrm{a} = \left|15\mathrm{a}\right|-3\mathrm{a} = 15\mathrm{a}-3\mathrm{a} = 12\mathrm{a} (\mathrm{vì} \mathrm{a}\ge 0).$

$\mathrm{d}) {\left(3-\mathrm{a}\right)}^2-\sqrt{0,2}.\sqrt{180{\mathrm{a}}^2} = {\left(3-\mathrm{a}\right)}^2-\sqrt{0,2.180{\mathrm{a}}^2} = {\left(3-\mathrm{a}\right)}^2-\sqrt{{\left(6\mathrm{a}\right)}^2} = 9-6\mathrm{a}+{\mathrm{a}}^2-\left|6\mathrm{a}\right|$

$$\begin{gathered} +) Nếu a \ge 0 \Rightarrow biểu thức trên =9-6\mathrm{a}+{\mathrm{a}}^2-6\mathrm{a} = 9-12\mathrm{a}+{\mathrm{a}}^2 \\ +) \mathrm{Nếu} \mathrm{a} < 0 \Rightarrow \mathrm{biểu} \mathrm{thức} \mathrm{trên} = 9-6\mathrm{a}+{\mathrm{a}}^2+6\mathrm{a} = 9 +\hspace{0.17em}{\mathrm{a}}^2 \end{gathered}$$