lên mạng mà tìm

Ta có:\(Q=\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{\sqrt{x}}{\sqrt{x}+1}+\frac{\sqrt{x}}{1-x}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{1-\sqrt{x}}{\sqrt{x}+1}\right)\)

             \(=\left(\frac{\left(\sqrt{x}+1\right)^2+\sqrt{x}\left(\sqrt{x}-1\right)+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\frac{\left(\sqrt{x}+1\right)^2-\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

             \(=\frac{x+2\sqrt{x}+1+x-\sqrt{x}+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}.\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{x+2\sqrt{x}+1-x+2\sqrt{x}-1}\)

             \(=\frac{2x+2\sqrt{x}+1}{4\sqrt{x}}\)

\(\left(\frac{3x+\sqrt{9x}-3}{x+\sqrt{x}-2}+\frac{1}{\sqrt{x}-1}+\frac{1}{\sqrt{x}+2}\right):\frac{1}{x-1}\)

\(\frac{3x+3\sqrt{x}-3+\sqrt{x}+2+\sqrt{x}-1}{x+\sqrt{x}-2}:\frac{1}{x-1}\)

\(\frac{3x+5\sqrt{x}-2}{x+\sqrt{x}-2}.x-1\)

\(\frac{3x-\sqrt{x}+6\sqrt{x}-2}{\sqrt{x}+2}\)

\(\frac{\sqrt{x}\left(3\sqrt{x}-1\right)+2\left(3\sqrt{x}-1\right)}{\sqrt{x}+2}\)

\(M=3\sqrt{x}-1\)

Ta có:\(M=\left(\frac{3x+\sqrt{9x}-3}{x+\sqrt{x}-2}+\frac{1}{\sqrt{x}-1}+\frac{1}{\sqrt{x}+2}\right):\frac{1}{x-1}\)

              \(=\left(\frac{3x+3\sqrt{x}-3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\right):\frac{1}{x-1}\)

              \(=\frac{3x+3\sqrt{x}-3+\sqrt{x}+2+\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}:\frac{1}{x-1}\)

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

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

\(a,\left(3-\frac{\sqrt{5}\left(\sqrt{5}+\sqrt{3}\right)}{\sqrt{5}+\sqrt{3}}\right)\left(3+\frac{\sqrt{5}\left(\sqrt{5}-1\right)}{\sqrt{5}-1}\right)\)

\(\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)\)

\(3^2-\sqrt{5}^2\)

\(=4\)

\(b,\frac{6\sqrt{2}-2\sqrt{6}}{\sqrt{6}-\sqrt{2}}+\frac{3}{3-\sqrt{2}}\)

\(\frac{\sqrt{6}\sqrt{2}\left(\sqrt{6}-\sqrt{2}\right)}{\sqrt{6}-\sqrt{2}}+\frac{3}{3-\sqrt{2}}\)

\(\sqrt{12}+\frac{3}{3-\sqrt{2}}\)

\(\frac{6\sqrt{3}-2\sqrt{6}-3}{3-\sqrt{2}}\)

\(c,\frac{4+2\sqrt{3}}{4-\sqrt{15}}\frac{\left(\sqrt{5}-\sqrt{3}\right)}{2}-2\sqrt{3}\)

\(\frac{2+\sqrt{3}\left(\sqrt{5}-\sqrt{3}\right)}{4-\sqrt{15}}-2\sqrt{3}\)

\(\frac{2\sqrt{5}+\sqrt{15}-2\sqrt{3}-3}{4-\sqrt{15}}-2\sqrt{3}\)

\(\frac{2\sqrt{5}+\sqrt{15}-2\sqrt{3}-3+6\sqrt{5}-8\sqrt{3}}{4-\sqrt{15}}\)

\(\frac{8\sqrt{5}+\sqrt{15}-10\sqrt{3}-3}{4-\sqrt{15}}\)

\(\sqrt{49a^2}+3a\)

\(\sqrt{49}\sqrt{a^2}+3a\)

\(7a+3a\)

\(=10a\)

Đk: \(a^2>b^2\); b \(\ne\)0

Q = \(\frac{a}{\sqrt{a^2-b^2}}-\left(1+\frac{a}{\sqrt{a^2-b^2}}\right):\frac{b}{a-\sqrt{a^2-b^2}}\)

Q = \(\frac{a}{\sqrt{a^2-b^2}}-\frac{\sqrt{a^2-b^2}+a}{\sqrt{a^2-b^2}}\cdot\frac{a-\sqrt{a^2-b^2}}{b}\)

Q = \(\frac{a}{\sqrt{a^2-b^2}}-\frac{a^2-\left(a^2-b^2\right)}{b\sqrt{a^2-b^2}}\)

Q = \(\frac{ab-b^2}{b\sqrt{a^2-b^2}}=\frac{b\left(a-b\right)}{b\sqrt{\left(a-b\right)\left(a+b\right)}}=\frac{\sqrt{a-b}}{\sqrt{a+b}}\)

\(abc\le\left(\frac{a+b+c}{3}\right)^3\Leftrightarrow\sqrt[3]{abc}\le\frac{a+b+c}{3}\)

BĐT Cô- si 

đánh giá từ tbn sang tbc đấy bạn 

\(A=\frac{2x-3\sqrt{x}-2}{\sqrt{x}-2}\)

\(A=\frac{2x+\sqrt{x}-4\sqrt{x}-2}{\sqrt{x}-2}\)

\(A=\frac{\sqrt{x}\left(2\sqrt{x}+1\right)-2\left(2\sqrt{x}+1\right)}{\sqrt{x}-2}\)

\(A=\frac{\sqrt{x}-2\left(2\sqrt{x}+1\right)}{\sqrt{x}-2}\)

\(A=2\sqrt{x}+1\)