Tính \(\sqrt{\dfrac{\left(\sqrt{3}+1\right)^2}{4}}\) được kết quả là bao nhiêu ? \(\dfrac{\sqrt{3}+1}{2}\) \(\dfrac{\sqrt{3}+1}{4}\) \(\sqrt{3}+1\) \(-\dfrac{\sqrt{3}+1}{2}\) Hướng dẫn giải: \(\sqrt{\dfrac{\left(\sqrt{3}+1\right)^2}{4}}=\dfrac{\sqrt{\left(\sqrt{3}+1\right)^2}}{\sqrt{4}}=\dfrac{\left|\sqrt{3}+1\right|}{2}=\dfrac{\sqrt{3}+1}{2}\)
Rút gọn biểu thức \(\dfrac{y}{x}\sqrt{\dfrac{x^2}{y^4}}\) với \(x>0\). \(\dfrac{1}{y}\) \(-\dfrac{1}{y}\) \(\dfrac{1}{y^2}\) \(\dfrac{x}{y}\) Hướng dẫn giải: \(\dfrac{y}{x}\sqrt{\dfrac{x^2}{y^4}}=\dfrac{y}{x}.\dfrac{\sqrt{x^2}}{\sqrt{y^4}}=\dfrac{y}{x}.\dfrac{x}{y^2}=\dfrac{1}{y}\).
Tính \(\left(3\sqrt{18}+2\sqrt{50}-4\sqrt{72}\right):8\sqrt{2}\) \(-\dfrac{5}{8}\) \(-\dfrac{3}{4}\) \(\dfrac{5}{8}\) \(\dfrac{1}{4}\) Hướng dẫn giải: \(\left(3\sqrt{18}+2\sqrt{50}-4\sqrt{72}\right):8\sqrt{2}\)\(=\dfrac{3\sqrt{18}}{8\sqrt{2}}+\dfrac{2\sqrt{50}}{8\sqrt{2}}-\dfrac{4\sqrt{72}}{8\sqrt{2}}\)\(=\dfrac{3}{8}.\sqrt{\dfrac{18}{2}}+\dfrac{2}{8}.\sqrt{\dfrac{50}{2}}-\dfrac{4}{8}.\sqrt{\dfrac{72}{2}}\) \(=\dfrac{3}{8}.\sqrt{9}+\dfrac{2}{8}.\sqrt{25}-\dfrac{4}{8}.\sqrt{36}\)\(=\dfrac{3}{8}.3+\dfrac{2}{8}.5-\dfrac{4}{8}.6\)\(=\dfrac{9+10-24}{8}=-\dfrac{5}{8}\)
Rút gọn \(\left(-4\sqrt{20}+5\sqrt{500}-3\sqrt{45}\right):\sqrt{5}\) 33 -33 \(3\sqrt{11}\) \(11\sqrt{3}\) Hướng dẫn giải: \(\left(-4\sqrt{20}+5\sqrt{500}-3\sqrt{45}\right):\sqrt{5}\)\(=\dfrac{-4\sqrt{20}}{\sqrt{5}}+\dfrac{5\sqrt{500}}{\sqrt{5}}-\dfrac{3\sqrt{45}}{\sqrt{5}}\)\(=-4.\sqrt{\dfrac{20}{5}}+5\sqrt{\dfrac{500}{5}}-3\sqrt{\dfrac{45}{5}}\) \(=-4.\sqrt{4}+5\sqrt{100}-3\sqrt{9}\)\(=-4.2+5.10-3.3=33\)
Rút gọn biểu thức \(\dfrac{\sqrt{a-2\sqrt{ab}+b}}{\sqrt{\sqrt{a}-\sqrt{b}}}\) với \(a,b\ge0\) và \(a>b\) \(\sqrt{\sqrt{a}-\sqrt{b}}\) \(\sqrt{a}-\sqrt{b}\) \(a-b\) \(\sqrt{\sqrt{a}+\sqrt{b}}\) Hướng dẫn giải: \(\dfrac{\sqrt{a-2\sqrt{ab}+b}}{\sqrt{\sqrt{a}-\sqrt{b}}}\)\(=\dfrac{\sqrt{\left(\sqrt{a}-\sqrt{b}\right)^2}}{\sqrt{\sqrt{a}-\sqrt{b}}}\)\(=\sqrt{\dfrac{\left(\sqrt{a}-\sqrt{b}\right)^2}{\sqrt{a}-\sqrt{b}}}=\sqrt{\sqrt{a}-\sqrt{b}}\)
Biết \(\sqrt{2}.x=\sqrt{50}\). Tìm x. \(x=5\) \(x=\sqrt{5}\) \(x=10\) \(x=2\sqrt{5}\) Hướng dẫn giải: \(\sqrt{2}.x=\sqrt{50}\)\(\Leftrightarrow x=\dfrac{\sqrt{50}}{\sqrt{2}}\)\(\Leftrightarrow x=\sqrt{25}=5\).
Tìm x biết \(\sqrt{3}x+\sqrt{3}=\sqrt{12}+\sqrt{27}\). x = 2 x = 3 x = 4 x = 5 Hướng dẫn giải: \(\sqrt{3}x+\sqrt{3}=\sqrt{12}+\sqrt{27}\)\(\Leftrightarrow\sqrt{3}x=\sqrt{12}+\sqrt{27}-\sqrt{3}\)\(\Leftrightarrow x=\dfrac{\sqrt{12}+\sqrt{27}-\sqrt{3}}{\sqrt{3}}\) \(\Leftrightarrow x=\sqrt{\dfrac{12}{3}}+\sqrt{\dfrac{27}{3}}-\sqrt{\dfrac{3}{3}}\)\(\Leftrightarrow x=\sqrt{4}+\sqrt{9}-\sqrt{1}\)\(\Leftrightarrow x=2+3-1=4\).
Số \(\sqrt{108}\) bằng: \(6\sqrt{3}\) \(3\sqrt{6}\) \(2\sqrt{6}\) \(4\sqrt{6}\) Hướng dẫn giải: \(\sqrt{108}=\sqrt{36.3}=\sqrt{6^2.3}=6\sqrt{3}\)
