Rút gọn \(\dfrac{1}{2a-1}\sqrt{4a^2\left(4a^2-4a+1\right)}\) với \(a>\dfrac{1}{2}\) \(2a\) \(-2a\) \(2\sqrt{a}\) \(2a\left(2a-1\right)\) Hướng dẫn giải: \(\dfrac{1}{2a-1}\sqrt{4a^2\left(4a^2-4a+1\right)}\)\(=\dfrac{1}{2a-1}.\sqrt{4a^2}.\sqrt{\left(2a-1\right)^2}\)\(=\dfrac{2a.\left(2a-1\right)}{2a-1}=2a\)
Khử mẫu biểu thức lấy căn: \(\sqrt{\dfrac{4}{5}}\) \(\dfrac{2\sqrt{5}}{5}\) \(\dfrac{\sqrt{5}}{5}\) \(\sqrt{5}\) \(\dfrac{3\sqrt{5}}{5}\) Hướng dẫn giải: \(\sqrt{\dfrac{4}{5}}\)\(=\sqrt{\dfrac{4.5}{5^2}}=\sqrt{\dfrac{2^2.5}{5^2}}=\dfrac{2\sqrt{5}}{5}\)
Khử mẫu biểu thức lấy căn số \(\sqrt{\dfrac{3}{4a^3}}\) \(\left(a>0\right)\) \(\dfrac{\sqrt{3a}}{2a^2}\) \(\dfrac{\sqrt{3a}}{2a}\) \(\dfrac{\sqrt{3}}{a^2}\) \(\dfrac{\sqrt{3}}{2a}\) Hướng dẫn giải: \(\sqrt{\dfrac{3}{4a^3}}\)\(=\sqrt{\dfrac{3.a}{4.a^3.a}}=\sqrt{\dfrac{3a}{4.a^4}}=\dfrac{\sqrt{3a}}{2a^2}\)
Trục căn thức ở mẫu : \(\dfrac{4}{\sqrt{5}-3}\) \(-3-\sqrt{5}\) \(3+\sqrt{5}\) \(\dfrac{4}{3+\sqrt{5}}\) \(3\sqrt{5}\) Hướng dẫn giải: \(\dfrac{4}{\sqrt{5}-3}=\dfrac{4.\left(\sqrt{5}+3\right)}{\left(\sqrt{5}-3\right)\left(\sqrt{5}+3\right)}\) \(=\dfrac{4.\left(\sqrt{5}+3\right)}{5-3^2}=\dfrac{4\left(\sqrt{5}+3\right)}{-4}\) \(=-3-\sqrt{5}\)
Rút gọn \(\dfrac{\sqrt{6}+\sqrt{10}}{\sqrt{21}+\sqrt{35}}\) \(\dfrac{\sqrt{14}}{7}\) \(\dfrac{2\sqrt{7}}{7}\) \(\dfrac{\sqrt{7}}{7}\) \(\dfrac{3\sqrt{7}}{7}\) Hướng dẫn giải: \(\dfrac{\sqrt{6}+\sqrt{10}}{\sqrt{21}+\sqrt{35}}=\dfrac{\sqrt{2}\left(\sqrt{3}+\sqrt{5}\right)}{\sqrt{7}\left(\sqrt{3}+\sqrt{5}\right)}\) \(=\dfrac{\sqrt{2}}{\sqrt{7}}=\dfrac{\sqrt{2}.\sqrt{7}}{\sqrt{7}.\sqrt{7}}=\dfrac{\sqrt{14}}{7}\)
Rút gọn biểu thức : \(\dfrac{\sqrt{6-2\sqrt{5}}}{\sqrt{5}-1}\) \(\sqrt{5}-1\) \(\sqrt{5}+1\) \(\left(\sqrt{5}-1\right)^2\) \(1\) Hướng dẫn giải: \(\dfrac{\sqrt{6-2\sqrt{5}}}{\sqrt{5}-1}=\dfrac{\sqrt{5-2\sqrt{5}+1}}{\sqrt{5}-1}\) \(=\dfrac{\sqrt{\left(\sqrt{5}-1\right)^2}}{\sqrt{5}-1}=\dfrac{\left|\sqrt{5}-1\right|}{\sqrt{5}-1}=1\)
Rút gọn: \(\left(\sqrt{125}+\sqrt{245}-\sqrt{5}\right):\sqrt{5}\) 11 10 12 9 Hướng dẫn giải: \(\left(\sqrt{125}+\sqrt{245}-\sqrt{5}\right):\sqrt{5}\) \(=\dfrac{\sqrt{125}+\sqrt{245}-\sqrt{5}}{\sqrt{5}}\) \(=\sqrt{\dfrac{125}{5}}+\sqrt{\dfrac{245}{5}}-\sqrt{\dfrac{5}{5}}\) \(=\sqrt{25}+\sqrt{49}-1=5+7-1=11\)
Rút gọn biểu thức : \(\dfrac{3-\sqrt{7}}{3+\sqrt{7}}+\dfrac{3+\sqrt{7}}{3-\sqrt{7}}\) 16 \(3+2\sqrt{7}\) \(3-\sqrt{7}\) \(6+4\sqrt{7}\) Hướng dẫn giải: \(\dfrac{3-\sqrt{7}}{3+\sqrt{7}}+\dfrac{3+\sqrt{7}}{3-\sqrt{7}}\)\(=\dfrac{\left(3-\sqrt{7}\right)^2+\left(3+\sqrt{7}\right)^2}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}\) \(=\dfrac{9-6\sqrt{7}+7+9+6\sqrt{7}+7}{9-7}\) \(=16\)
Rút gọn biểu thức: \(\dfrac{5}{12\left(2\sqrt{5}+3\sqrt{2}\right)}-\dfrac{5}{12\left(2\sqrt{5}-3\sqrt{2}\right)}\) \(-\dfrac{5\sqrt{2}}{4}\) \(\dfrac{5\sqrt{2}}{4}\) \(\dfrac{\sqrt{2}}{4}\) \(-\dfrac{\sqrt{2}}{4}\) Hướng dẫn giải: \(\dfrac{5}{12\left(2\sqrt{5}+3\sqrt{2}\right)}-\dfrac{5}{12\left(2\sqrt{5}-3\sqrt{2}\right)}\) \(=\dfrac{5\left(2\sqrt{5}-3\sqrt{2}\right)}{12\left(2\sqrt{5}+3\sqrt{2}\right)\left(2\sqrt{5}-3\sqrt{2}\right)}-\dfrac{5\left(2\sqrt{5}+3\sqrt{2}\right)}{12\left(2\sqrt{5}+3\sqrt{2}\right)\left(2\sqrt{5}-3\sqrt{2}\right)}\) \(=\dfrac{10\sqrt{5}-15\sqrt{2}-10\sqrt{5}-15\sqrt{2}}{12.\left(20-18\right)}\) \(=\dfrac{-30\sqrt{2}}{24}=\dfrac{-5\sqrt{2}}{4}\)
Rút gọn biểu thức: \(\sqrt{20}+2\sqrt{45}-\sqrt{1\dfrac{1}{4}}-\sqrt{5}\) \(\dfrac{13\sqrt{5}}{2}\) \(-\dfrac{13\sqrt{5}}{2}\) \(5\sqrt{5}\) \(\dfrac{9\sqrt{5}}{2}\) Hướng dẫn giải: \(\sqrt{20}+2\sqrt{45}-\sqrt{1\dfrac{1}{4}}-\sqrt{5}\) \(=2\sqrt{5}+2.3\sqrt{5}-\sqrt{\dfrac{5}{4}}-\sqrt{5}\) \(=7\sqrt{5}-\dfrac{\sqrt{5}}{2}\) \(=\dfrac{13\sqrt{5}}{2}\)
