Đáp án:
\(a)Q=\left(\dfrac{1}{\sqrt{x}+2}+\dfrac{7}{x-4}\right):\left(\dfrac{\sqrt{x}-1}{\sqrt{x}-2}-1\right)(x \ge 0,x \ne 4)\)
\(Q=\left(\dfrac{1}{\sqrt{x}+2}+\dfrac{7}{(\sqrt{x}-2)(\sqrt{x}+2)}\right):\left(\dfrac{\sqrt{x}-1}{\sqrt{x}-2}-1\right)\)
\(Q=\left(\dfrac{\sqrt{x}-2}{(\sqrt{x}-2)(\sqrt{x}+2)}+\dfrac{7}{(\sqrt{x}-2)(\sqrt{x}+2)}\right):\left(\dfrac{(\sqrt{x}-1)(\sqrt{x}+2)}{(\sqrt{x}-2)(\sqrt{x}+2)}-\dfrac{(\sqrt{x}-2)(\sqrt{x}+2)}{(\sqrt{x}-2)(\sqrt{x}+2)}\right)\)
\(Q=\left(\dfrac{\sqrt{x}-2+7}{(\sqrt{x}-2)(\sqrt{x}+2)}\right):\dfrac{x+\sqrt{x}-2-x+4}{(\sqrt{x}-2)(\sqrt{x}+2)}\)
\(Q=\dfrac{\sqrt{x}+5}{(\sqrt{x}-2)(\sqrt{x}+2)}:\dfrac{\sqrt{x}+2}{(\sqrt{x}-2)(\sqrt{x}+2)}\)
\(Q=\dfrac{\sqrt{x}+5}{\sqrt{x}+2}\)
Vậy giá trị rút gọn của Q là \(Q=\dfrac{\sqrt{x}+5}{\sqrt{x}+2}\)
\(b_1)x=\sqrt{27+10\sqrt{2}}-\sqrt{18+8\sqrt{2}}\)
\(x=\sqrt{25+2.5.\sqrt{2}+2}-\sqrt{16+2.4.\sqrt{2}+2}\)
\(x=\sqrt{(5+\sqrt{2})^2}-\sqrt{(4+\sqrt{2})^2}\)
\(x=|5+\sqrt{2}|-|4+\sqrt{2}|\)
\(x=5+\sqrt{2}-4-\sqrt{2}=1\)
\(\Rightarrow Q=\dfrac{1+5}{1+2}=\dfrac{6}{3}=2\)
\(b_2)x=\sqrt{\dfrac{2}{2-\sqrt{3}}}-\sqrt{\dfrac{2}{2+\sqrt{3}}}\)
\(x=\sqrt{\dfrac{4}{4-2\sqrt{3}}}-\sqrt{\dfrac{4}{4+2\sqrt{3}}}\)
\(x=\sqrt{\dfrac{4}{3-2\sqrt{3}+1}}-\sqrt{\dfrac{4}{3+2\sqrt{3}+1}}\)
\(x=\sqrt{\dfrac{4}{(\sqrt{3}-1)^2}}-\sqrt{\dfrac{4}{(\sqrt{3}+1)^2}}\)
\(x=\left|\dfrac{2}{\sqrt{3}-1}\right|-\left|\dfrac{2}{\sqrt{3}+1}\right|\)
\(x=\dfrac{2(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}-\dfrac{2(\sqrt{3}-1)}{(\sqrt{3}-1)(\sqrt{3}+1)}\)
\(x=\dfrac{2(\sqrt{3}+1)}{3-1}-\dfrac{2(\sqrt{3}-1)}{3-1}\)
\(x=\sqrt{3}+1-\sqrt{3}+1=2\)
\(\Rightarrow Q=\dfrac{\sqrt{2}+5}{\sqrt{2}+2}\)
\(Q=\dfrac{(5+\sqrt{2})(2-\sqrt{2})}{(2+\sqrt{2})(2-\sqrt{2}}\)
\(Q=\dfrac{10-5\sqrt{2}+2\sqrt{2}-2}{4-2}\)
\(Q=\dfrac{8-3\sqrt{2}}{2}\)
Vậy với \(x=\sqrt{27+10\sqrt{2}}-\sqrt{18+8\sqrt{2}}\) thì \(Q=2\) và với \(x=\sqrt{\dfrac{2}{2-\sqrt{3}}}-\sqrt{\dfrac{2}{2+\sqrt{3}}}\) thì \(Q=\dfrac{8-3\sqrt{2}}{2}\).
