Rút gọn biểu thức: \(\sqrt{28}+\sqrt{\dfrac{7}{2}.0,32}+\sqrt{35}.\sqrt{5}-\sqrt{7}\) \(\dfrac{32\sqrt{7}}{5}\) \(-\dfrac{32\sqrt{7}}{5}\) \(6\sqrt{7}\) \(\dfrac{33\sqrt{7}}{5}\) Hướng dẫn giải: \(\sqrt{28}+\sqrt{\dfrac{7}{2}.0,32}+\sqrt{35}.\sqrt{5}-\sqrt{7}\) \(=2\sqrt{7}+\sqrt{7.0,16}+\sqrt{7.5.5}-\sqrt{7}\) \(=2\sqrt{7}+\dfrac{2\sqrt{7}}{5}+5\sqrt{7}-\sqrt{7}\) \(=\dfrac{32\sqrt{7}}{5}\)
Rút gọn biểu thức \(\left(\sqrt{28}-2\sqrt{3}+\sqrt{7}\right).\sqrt{7}+\sqrt{84}\) \(21+\sqrt{7}\) 21 \(20+2\sqrt{7}\) \(18+\sqrt{7}\) Hướng dẫn giải: \(\left(\sqrt{28}-2\sqrt{3}+\sqrt{7}\right).\sqrt{7}+\sqrt{84}\) \(=\left(2\sqrt{7}-2\sqrt{3}+\sqrt{7}\right).\sqrt{7}+2\sqrt{21}\) \(=2.7-2\sqrt{21}+7+2\sqrt{21}=21\).
Rút gọn biểu thức \(\dfrac{a-b}{\sqrt{a}-\sqrt{b}}+\dfrac{\sqrt{a}^3+\sqrt{b}^3}{a-b}\) \(\dfrac{2a-\sqrt{ab}}{\sqrt{a}-\sqrt{b}}\) \(\sqrt{a}\) \(2\sqrt{a}\) \(\sqrt{a}+\sqrt{b}\) Hướng dẫn giải: \(\dfrac{a-b}{\sqrt{a}-\sqrt{b}}+\dfrac{\sqrt{a}^3+\sqrt{b}^3}{a-b}\) \(=\sqrt{a}+\sqrt{b}+\dfrac{\left(\sqrt{a}+\sqrt{b}\right)\left(a-\sqrt{ab}+b\right)}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}\) \(=\sqrt{a}+\sqrt{b}+\dfrac{a-\sqrt{ab}+b}{\sqrt{a}-\sqrt{b}}\) \(=\dfrac{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{a}-\sqrt{b}}+\dfrac{a-\sqrt{ab}+b}{\sqrt{a}-\sqrt{b}}\) \(=\dfrac{a-b+a-\sqrt{ab}+b}{\sqrt{a}-\sqrt{b}}\) \(=\dfrac{2a-\sqrt{ab}}{\sqrt{a}-\sqrt{b}}\)
Rút gọn biểu thức \(\left(2-\sqrt{2}\right)\left(-5\sqrt{2}\right)-\left(3\sqrt{2}-5\right)^2\) \(-33+20\sqrt{2}\) \(-30+20\sqrt{2}\) \(-40+20\sqrt{2}\) \(-33+30\sqrt{2}\) Hướng dẫn giải: \(\left(2-\sqrt{2}\right)\left(-5\sqrt{2}\right)-\left(3\sqrt{2}-5\right)^2\) \(=-10\sqrt{2}-5\sqrt{2}.\sqrt{2}-\left(18-30\sqrt{2}+25\right)\) \(=-10\sqrt{2}-5.2-18+30\sqrt{2}-25\) \(=-33+20\sqrt{2}\)
Rút gọn biểu thức \(2\sqrt{3a}-\sqrt{75a}+a\sqrt{\dfrac{6}{5}.\dfrac{5}{2a}}-\dfrac{2}{5}\sqrt{300a^3}\) (a > 0) \(-2\sqrt{3a}\left(1+2a\right)\) \(2\sqrt{3a}\left(1+2a\right)\) \(-2\sqrt{3a}\left(1-2a\right)\) \(-2\sqrt{3a}\) Hướng dẫn giải: \(2\sqrt{3a}-\sqrt{75a}+a\sqrt{\dfrac{6}{5}.\dfrac{5}{2a}}-\dfrac{2}{5}\sqrt{300a^3}\) \(=2\sqrt{3a}-5\sqrt{3a}+a\sqrt{\dfrac{3}{a}}-\dfrac{2}{5}.10.a\sqrt{3a}\) \(=-3\sqrt{3a}+\sqrt{\dfrac{3}{a}.a^2}-4a\sqrt{3a}\) \(=-3\sqrt{3a}+\sqrt{3a}-4a\sqrt{3a}\) \(=-2\sqrt{3a}-4a\sqrt{3a}\) \(=-2\sqrt{3a}\left(1+2a\right)\).
Giải phương trình : \(\sqrt{36x+72}-\sqrt{\dfrac{1}{4}x+\dfrac{1}{2}}+\dfrac{3}{2}\sqrt{x+2}=14\). \(x=2\) \(x=1\) \(x=\dfrac{1}{2}\) \(x=3\) Hướng dẫn giải: Đkxđ: \(x\ge-2\). \(\sqrt{36x+72}-\sqrt{\dfrac{1}{4}x+\dfrac{1}{2}}+\dfrac{3}{2}\sqrt{x+2}=14\) \(\Leftrightarrow\sqrt{36\left(x+2\right)}-\sqrt{\dfrac{1}{4}\left(x+2\right)}+\dfrac{3}{2}\sqrt{x+2}=14\) \(\Leftrightarrow6\sqrt{x+2}-\dfrac{1}{2}\sqrt{x+2}+\dfrac{3}{2}\sqrt{x+2}=14\) \(\Leftrightarrow7\sqrt{x+2}=14\) \(\Leftrightarrow\sqrt{x+2}=2\) \(\Leftrightarrow x+2=4\) \(\Leftrightarrow x=2\)
Rút gọn biểu thức \(\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-1}-\dfrac{\sqrt{a}-1}{\sqrt{a}+1}+4\sqrt{a}\right)\left(\sqrt{a}-\dfrac{1}{\sqrt{a}}\right)\) \(4a\) \(-4a\) \(a+1\) \(a-1\) Hướng dẫn giải: \(\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-1}-\dfrac{\sqrt{a}-1}{\sqrt{a}+1}+4\sqrt{a}\right)\left(\sqrt{a}-\dfrac{1}{\sqrt{a}}\right)\) \(=\left(\dfrac{\left(\sqrt{a}+1\right)^2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}-\dfrac{\left(\sqrt{a}-1\right)^2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+4\sqrt{a}\right)\left(\dfrac{a-1}{\sqrt{a}}\right)\) \(=\left(\dfrac{a+2\sqrt{a}+1-\left(a-2\sqrt{a}+1\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+4\sqrt{a}\right).\dfrac{a-1}{\sqrt{a}}\) \(=\left(\dfrac{4\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+4\sqrt{a}\right).\dfrac{a-1}{\sqrt{a}}\) \(=4\sqrt{a}\left(\dfrac{1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+1\right).\dfrac{a-1}{\sqrt{a}}\) \(=4\left(\dfrac{1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right).\left(a-1\right)\) \(=4\left[1+\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)\right]\) \(=4a\)
Biết \(\dfrac{x}{x-4}+\dfrac{1}{\sqrt{x}-2}+\dfrac{1}{\sqrt{x}+2}=2\). Tìm x. \(x=8\) \(x=\dfrac{1}{4}\) \(x=2\sqrt{2}\) \(x=16\) Hướng dẫn giải: Đkxđ: \(x\ne4\) \(\dfrac{x}{x-4}+\dfrac{1}{\sqrt{x}-2}+\dfrac{1}{\sqrt{x}+2}=2\) \(\Leftrightarrow\dfrac{x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\dfrac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\dfrac{\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=2\) \(\Leftrightarrow\dfrac{x+2\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=2\) \(\Leftrightarrow\dfrac{\sqrt{x}\cdot\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=2\) \(\Leftrightarrow\dfrac{\sqrt{x}}{\sqrt{x}-2}=2\) \(\Leftrightarrow\sqrt{x}=2\left(\sqrt{x}-2\right)\) \(\Leftrightarrow\sqrt{x}=4\) \(\Leftrightarrow x=16\).
Tính giá trị của biểu thức \(\dfrac{\sqrt{3-\sqrt{5}}\left(3+\sqrt{5}\right)}{\sqrt{10}+\sqrt{2}}\) 1 \(\sqrt{2}\) \(\sqrt{3}\) 2 Hướng dẫn giải: \(\dfrac{\sqrt{3-\sqrt{5}}\left(3+\sqrt{5}\right)}{\sqrt{10}+\sqrt{2}}\) \(=\dfrac{\sqrt{6-2\sqrt{5}}\left(3+\sqrt{5}\right)}{\sqrt{20}+2}\) \(=\dfrac{\sqrt{\left(\sqrt{5}-1\right)^2}\left(3+\sqrt{5}\right)}{2\left(\sqrt{5}+1\right)}\) \(=\dfrac{\left(\sqrt{5}-1\right).\left(3+\sqrt{5}\right)}{2\left(\sqrt{5}+1\right)}\) \(=\dfrac{3\sqrt{5}+5-3-\sqrt{5}}{2\left(\sqrt{5}+1\right)}\) \(=\dfrac{2+2\sqrt{5}}{2\left(\sqrt{5}+1\right)}=1\)