\(\sqrt{1+x^2+\frac{x^2}{\left(x+1\right)^2}}+\frac{1}{x+1}\)
\(=\sqrt{x^2+2x+1-2x+\frac{x^2}{\left(x+1\right)^2}}+\frac{1}{x+1}\)
\(=\sqrt{\left(x+1\right)^2-2x+\left(\frac{x}{x+1}\right)^2}+\frac{1}{x+1}\)
\(=\sqrt{\left(x+1-\frac{x}{x+1}\right)^2}+\frac{1}{x+1}\)
\(=x+1-\frac{x}{x+1}+\frac{1}{x+1}\)
\(=\frac{x^2+2x+1-x+1}{x+1}\)
\(=\frac{x^2+x+2}{x+1}\)
1. Rút gọn
P=\(2\sqrt{1+\frac{1}{4}\left(\sqrt{\frac{1}{x}}-\sqrt{x}\right)^2}:\left[\sqrt{1+\frac{1}{4}\left(\sqrt{\frac{1}{x}}-\sqrt{x}\right)^2}-\frac{1}{2}\left(\sqrt{\frac{1}{x}}-\sqrt{x}\right)^2\right]\)
Rút gọn:
\(\left(\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}—2}-\frac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)
Rút gọn A = \(1-\left(\frac{2x-1+\sqrt{x}}{1-x}+\frac{2x\sqrt{x}+x-\sqrt{x}}{1+x\sqrt{x}}\right).\left(\frac{\left(x-\sqrt{x}\right)\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}\right)\)
Rút gọn A = \(\left(\frac{x+2\sqrt{x}+4}{x\sqrt{x}-8}+\frac{x+2\sqrt{x}+1}{x-1}\right) :\left(3+\frac{1}{\sqrt{x}-2}+\frac{2}{\sqrt{x}+1}\right)\)
a, Rút gọn A b , Tìm x thỏa mãn A > 1 c,Tính A với \(x=\frac{\sqrt{4+\sqrt{3}}+\sqrt{4-\sqrt{3}}}{\sqrt{4+\sqrt{13}}}+\sqrt{27-10\sqrt{2}}\)\(A=\frac{\sqrt{x}+1}{3\left(\sqrt{x}-1\right)}\)
Rút gọn: \(P=1-\left(\frac{2x-1+\sqrt{x}}{1-x}+\frac{2x\sqrt{x}+x-\sqrt{x}}{1+x\sqrt{x}}\right).\frac{\left(x-\sqrt{x}\right)\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}\)
Rút gọn \(M=\left(\frac{2x+3\sqrt{x}}{2\sqrt{x}+1}+\frac{1}{x-\sqrt{x}+1}-\frac{1}{\sqrt{x}+1}\right).\left(\frac{x-\sqrt{x}+1}{\sqrt{x}}\right)\)
rút gọn \(\left(\frac{2x+1}{x\sqrt{x}-1}+\frac{1}{1-\sqrt{x}}\right):\left(1-\frac{x-2}{x\sqrt{x}+1}\right)\)
\(A=\left(\frac{1}{1-\sqrt{x}}+\frac{1}{1+\sqrt{x}}\right):\left(\frac{1}{1-\sqrt{x}}-\frac{1}{1+\sqrt{x}}\right)+\frac{1}{2\sqrt{x}}\)
a)rút gọn A
\(P=\frac{x\sqrt{x}-1}{x-\sqrt{x}}-\frac{x\sqrt{x}+1}{x+\sqrt{x}}+\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)\left(\frac{\sqrt{x}+1}{\sqrt{x-1}}+\frac{\sqrt{x}-1}{\sqrt{x}+1}\right)-2\sqrt{x}\)
rút gọn P
Rút gọn các biểu thức sau:
a) A=\(\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{3}-1}+\frac{2\sqrt{2}-\sqrt{6}}{\sqrt{2}}\)
b)B=\(\left(\frac{1}{x-\sqrt{x}}+\frac{1}{\sqrt{x}-1}\right):\frac{\sqrt{x}}{x-2\sqrt{x}+1}\left(x\ge0;x\ne1\right)\)
