a) \(\left(\dfrac{\sqrt{6}-\sqrt{2}}{1-\sqrt{3}}-\dfrac{\sqrt{5}-5}{1-\sqrt{5}}\right):\dfrac{1}{\sqrt{2}-\sqrt{5}}\)

\(=\left[-\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)}{\sqrt{3}-1}-\dfrac{\sqrt{5}\left(1-\sqrt{5}\right)}{1-\sqrt{5}}\right]\cdot\left(\sqrt{2}-\sqrt{5}\right)\)

\(=\left(-\sqrt{2}-\sqrt{5}\right)\left(\sqrt{2}-\sqrt{5}\right)\)

\(=-\left(\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{5}\right)\)

\(=-\left(2-5\right)\)

\(=-\left(-3\right)\)

\(=3\)

b) Ta có:

\(x^2-x\sqrt{3}+1\) 

\(=x^2-2\cdot\dfrac{\sqrt{3}}{2}\cdot x+\left(\dfrac{\sqrt{3}}{2}\right)^2+\dfrac{1}{4}\)

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

Mà: \(\left(x-\dfrac{\sqrt{3}}{2}\right)^2\ge0\forall x\) nên

\(\left(x-\dfrac{\sqrt{3}}{2}\right)^2+\dfrac{1}{4}\ge\dfrac{1}{4}\forall x\)

Dấu "=" xảy ra:

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

\(\Leftrightarrow x=\dfrac{\sqrt{3}}{2}\)

Vậy: GTNN của biểu thức là \(\dfrac{1}{4}\) tại \(x=\dfrac{\sqrt{3}}{2}\)

a)

\(\left(\dfrac{\sqrt{6}-\sqrt{2}}{1-\sqrt{3}}-\dfrac{\sqrt{5}-5}{1-\sqrt{5}}\right):\dfrac{1}{\sqrt{2}-\sqrt{5}}\\ =\left(-\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)}{\sqrt{3}-1}-\dfrac{\sqrt{5}\left(1-\sqrt{5}\right)}{1-\sqrt{5}}\right).\left(\sqrt{2}-\sqrt{5}\right)\\ =\left(-\sqrt{2}-\sqrt{5}\right).\left(\sqrt{2}-\sqrt{5}\right)\\ =-\left(\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{5}\right)\\ =-\left(\sqrt{2}^2-\sqrt{5}^2\right)\\ =-\left(2-5\right)\\ =-\left(-3\right)\\ =3\)

\(Q=\dfrac{2}{2+\sqrt{x}}+\dfrac{1}{2-\sqrt{x}}+\dfrac{2\sqrt{x}}{x-4}\left(dk:x\ge0,x\ne4\right)\\ =\dfrac{2}{2+\sqrt{x}}+\dfrac{1}{2-\sqrt{x}}-\dfrac{2\sqrt{x}}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\\ =\dfrac{2\left(2-\sqrt{x}\right)+2+\sqrt{x}-2\sqrt{x}}{4-x}\\ =\dfrac{4-2\sqrt{x}+2+\sqrt{x}-2\sqrt{x}}{4-x}\\ =\dfrac{-3\sqrt{x}+6}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\\ =\dfrac{-3\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\\ =\dfrac{3}{\sqrt{x}+2}\)

\(b,Q=\dfrac{6}{5}\Leftrightarrow\dfrac{3}{\sqrt{x}+2}=\dfrac{6}{5}\Rightarrow15-6\left(\sqrt{x}+2\right)=0\Rightarrow15-6\sqrt{x}-12=0\)

\(\Rightarrow-6\sqrt{x}=-3\Rightarrow\sqrt{x}=\dfrac{1}{2}\Rightarrow x=\dfrac{1}{4}\left(tm\right)\)

Vậy \(x=\dfrac{1}{4}\)thỏa mãn đề bài.

Mình ghi nhầm. \(x=\frac{\sqrt{4+2\sqrt{3}}.\left(\sqrt{3}-1\right)}{\sqrt{6+2\sqrt{5}}-\sqrt{5}}\)nhé

1.052631148

có hiểu rút gọn là j ko thế

\(=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{\frac{\sqrt{2}\left(\sqrt{\sqrt{3}}+1\right)}{\sqrt{2}\left(\sqrt{\sqrt{3}}-1\right)}}=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{\frac{\left(\sqrt{\sqrt{3}}+1\right)^2}{\left(\sqrt{\sqrt{3}}-1\right)\left(\sqrt{\sqrt{3}}+1\right)}}\)\(=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{\frac{\sqrt{3}+1+2\sqrt{\sqrt{3}}}{\sqrt{3}-1}}\)\(=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{\frac{\left(\sqrt{3}+1+2\sqrt{\sqrt{3}}\right)\left(\sqrt{3}+1\right)}{2}}=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{2+\sqrt{3}+\sqrt{\sqrt{3}}+\sqrt{3\sqrt{3}}}\)

\(=\frac{\sqrt{30}+\sqrt{5}}{\left(\sqrt{3}+1\right)\left(\sqrt{\sqrt{3}}+1\right)+1}=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{\sqrt{3}}+1\right)\left(\sqrt{\sqrt{3}}-1\right)+\sqrt{\sqrt{3}}-1}\)

\(=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)+\sqrt{\sqrt{3}}-1}\)

\(=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)\left(\sqrt{\sqrt{3}}-1\right)}{\left(\sqrt{\sqrt{3}}+1\right)\left(\sqrt{\sqrt{3}}-1\right)}\)

\(=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)^2\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\)

\(=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)^2\left(\sqrt{3}+1\right)}{2}\)\(=2\sqrt{30}+2\sqrt{5}+\sqrt{90}+\sqrt{15}-\sqrt{90\sqrt{3}}-\sqrt{30\sqrt{3}}-\sqrt{15\sqrt{3}}-\sqrt{5\sqrt{3}}\)

mởi tay ùi,có gì thiếu tự giải tiếp ^^

Với mọi \(k\ge2\)  thì \(\frac{2k+\sqrt{k^2-1}}{\sqrt{k-1}+\sqrt{k+1}}=\frac{\left[\left(\sqrt{k-1}\right)^2+\left(\sqrt{k+1}\right)^2+\sqrt{\left(k-1\right)\left(k+1\right)}\right]\left(\sqrt{k+1}-\sqrt{k-1}\right)}{\left(\sqrt{k-1}+\sqrt{k+1}\right)\left(\sqrt{k+1}-\sqrt{k-1}\right)}\)

                                                \(=\frac{\sqrt{\left(k+1\right)^3}-\sqrt{\left(k-1\right)^3}}{2}\)

Suy ra tổng đã cho có thể viết là :

\(A=\frac{1}{2}\left[\sqrt{3^3}-\sqrt{1^3}+\sqrt{4^3}-\sqrt{2^3}+\sqrt{5^3}-\sqrt{3^3}+\sqrt{6^3}-\sqrt{4^3}+...+\sqrt{101^3}-\sqrt{99^3}\right]\)

    \(=\frac{1}{2}\left[-1-\sqrt{2^3}+\sqrt{101^3}+\sqrt{100^3}\right]\)

   \(=\frac{999+\sqrt{101^3}-\sqrt{8}}{2}\)

a) \(=\frac{7-4\sqrt{3}+7+4\sqrt{3}}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}=\frac{14}{49-48}=14\)

b) \(=\frac{15\left(\sqrt{6}-1\right)}{\left(\sqrt{6}+1\right)\left(\sqrt{6}-1\right)}-\frac{5\sqrt{6}}{5}+\frac{4\sqrt{3}-12\sqrt{2}}{\sqrt{6}\left(\sqrt{3}-\sqrt{2}\right)}\)

\(D=\sqrt{\frac{\left(5+2\sqrt{6}\right)^2}{25-24}}+\sqrt{\frac{\left(5-2\sqrt{6}\right)^2}{25-24}}=5+2\sqrt{6}+5-2\sqrt{6}=10\)