Liên hệ giữa phép chia và phép khai phương

​\(\dfrac{\sqrt{0,4}}{\sqrt{2,5}}=\sqrt{\dfrac{0,4}{2,5}}=\sqrt{\dfrac{4}{25}}=\sqrt{\left(\dfrac{2}{5}\right)^2}=\dfrac{2}{5}\)

​\(\dfrac{\sqrt{0,001}}{\sqrt{0,004}}=\sqrt{\dfrac{0,001}{0,004}}=\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}\)

​\(\sqrt{\dfrac{\left(\sqrt{3}+1\right)^2}{4}}=\dfrac{\sqrt{\left(\sqrt{3}+1\right)^2}}{\sqrt{4}}=\dfrac{\left|\sqrt{3}+1\right|}{2}=\dfrac{\sqrt{3}+1}{2}\)

​\(\dfrac{y}{x}\sqrt{\dfrac{x^2}{y^4}}=\dfrac{y}{x}.\dfrac{\sqrt{x^2}}{\sqrt{y^4}}=\dfrac{y}{x}.\dfrac{x}{y^2}=\dfrac{1}{y}\).

​\(\left(3\sqrt{18}+2\sqrt{50}-4\sqrt{72}\right):8\sqrt{2}\)\(=\dfrac{3\sqrt{18}}{8\sqrt{2}}+\dfrac{2\sqrt{50}}{8\sqrt{2}}-\dfrac{4\sqrt{72}}{8\sqrt{2}}\)\(=\dfrac{3}{8}.\sqrt{\dfrac{18}{2}}+\dfrac{2}{8}.\sqrt{\dfrac{50}{2}}-\dfrac{4}{8}.\sqrt{\dfrac{72}{2}}\)
\(=\dfrac{3}{8}.\sqrt{9}+\dfrac{2}{8}.\sqrt{25}-\dfrac{4}{8}.\sqrt{36}\)\(=\dfrac{3}{8}.3+\dfrac{2}{8}.5-\dfrac{4}{8}.6\)\(=\dfrac{9+10-24}{8}=-\dfrac{5}{8}\)
 

​\(\left(-4\sqrt{20}+5\sqrt{500}-3\sqrt{45}\right):\sqrt{5}\)\(=\dfrac{-4\sqrt{20}}{\sqrt{5}}+\dfrac{5\sqrt{500}}{\sqrt{5}}-\dfrac{3\sqrt{45}}{\sqrt{5}}\)\(=-4.\sqrt{\dfrac{20}{5}}+5\sqrt{\dfrac{500}{5}}-3\sqrt{\dfrac{45}{5}}\)
\(=-4.\sqrt{4}+5\sqrt{100}-3\sqrt{9}\)\(=-4.2+5.10-3.3=33\)

​\(\dfrac{\sqrt{a-2\sqrt{ab}+b}}{\sqrt{\sqrt{a}-\sqrt{b}}}\)\(=\dfrac{\sqrt{\left(\sqrt{a}-\sqrt{b}\right)^2}}{\sqrt{\sqrt{a}-\sqrt{b}}}\)\(=\sqrt{\dfrac{\left(\sqrt{a}-\sqrt{b}\right)^2}{\sqrt{a}-\sqrt{b}}}=\sqrt{\sqrt{a}-\sqrt{b}}\)

​\(\sqrt{2}.x=\sqrt{50}\)\(\Leftrightarrow x=\dfrac{\sqrt{50}}{\sqrt{2}}\)\(\Leftrightarrow x=\sqrt{25}=5\).

​\(\sqrt{3}x+\sqrt{3}=\sqrt{12}+\sqrt{27}\)\(\Leftrightarrow\sqrt{3}x=\sqrt{12}+\sqrt{27}-\sqrt{3}\)\(\Leftrightarrow x=\dfrac{\sqrt{12}+\sqrt{27}-\sqrt{3}}{\sqrt{3}}\)
\(\Leftrightarrow x=\sqrt{\dfrac{12}{3}}+\sqrt{\dfrac{27}{3}}-\sqrt{\dfrac{3}{3}}\)\(\Leftrightarrow x=\sqrt{4}+\sqrt{9}-\sqrt{1}\)\(\Leftrightarrow x=2+3-1=4\).