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a, dk \(x\ge0.x\ne1\)
\(\left(\frac{1+\sqrt{x}+1-\sqrt{x}}{2\left(1-x\right)}-\frac{x^2+1}{1-x^2}\right)\left(\frac{x+1}{x}\right)\)=\(\left(\frac{1}{1-x}-\frac{x^2+1}{1-x^2}\right)\left(\frac{x+1}{x}\right)\)
=\(\left(\frac{1+x-x^2-1}{1-x^2}\right)\left(\frac{x+1}{x}\right)=\frac{x\left(1-x\right)\left(x+1\right)}{x\left(1-x\right)\left(1+x\right)}=1\)
phan b,c ban tu lam not nhe dai lam mk ko lam dau mk co vc ban rui
Bài 1:
a) \(\frac{2}{\sqrt{3}-1}-\frac{2}{\sqrt{3}+1}\)
\(=\frac{2\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}-\frac{2\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}\)
\(=\frac{2\left(\sqrt{3}+1\right)}{2}-\frac{2\left(\sqrt{3}-1\right)}{2}\)
\(=\sqrt{3}+1-\left(\sqrt{3}-1\right)=2\)
b) \(\frac{2}{5-\sqrt{3}}+\frac{3}{\sqrt{6}+\sqrt{3}}\)
\(=\frac{2\left(5+\sqrt{3}\right)}{\left(5-\sqrt{3}\right)\left(5+\sqrt{3}\right)}+\frac{3\left(\sqrt{6}-\sqrt{3}\right)}{\left(\sqrt{6}+\sqrt{3}\right)\left(\sqrt{6}-\sqrt{3}\right)}\)
\(=\frac{2\left(5+\sqrt{3}\right)}{2}+\frac{3\left(\sqrt{6}-\sqrt{3}\right)}{3}\)
\(=5+\sqrt{3}+\sqrt{6}-\sqrt{3}=5+\sqrt{6}\)
c) ĐK: \(a\ge0;a\ne1\)
\(\left(1+\frac{a+\sqrt{a}}{1+\sqrt{a}}\right).\left(1-\frac{a-\sqrt{a}}{\sqrt{a}-1}\right)+a\)
\(=\left(1+\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{1+\sqrt{a}}\right).\left(1-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)+a\)
\(=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)+a\)
\(=1-a+a=1\)
Ta có: \(A=\left(\frac{1}{\sqrt{x-1}}+\frac{1}{\sqrt{x+1}}\right)^2\times\frac{x^2-1}{2}-\sqrt{1-x^2}\) \(\left(ĐK:x\ge1\right)\)
\(\Leftrightarrow A=\left(\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x-1}.\sqrt{x+1}}\right)^2\times\frac{x^2-1}{2}-\sqrt{1-x^2}\)
\(\Leftrightarrow A=\frac{x+1+x-1+2\sqrt{x^2-1}}{x^2-1}\times\frac{x^2-1}{2}-\sqrt{1-x^2}\)
\(\Leftrightarrow A=\frac{2x+2\sqrt{x^2-1}}{2}-\sqrt{1-x^2}\)
\(\Leftrightarrow A=x+\sqrt{1-x^2}-\sqrt{1-x^2}\)
\(\Leftrightarrow A=x\)
Học tốt
ĐKXĐ : ...............
\(A=\left(\frac{1}{\sqrt{x-1}}+\frac{1}{\sqrt{x+1}}\right)^2\times\frac{x^2-1}{2}-\sqrt{1-x^2}\)
\(A=\left(\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{\left(x-1\right)\left(x+1\right)}}\right)^2\times\frac{x^2-1}{2}-\sqrt{1-x^2}\)
\(A=\frac{\left(\sqrt{x+1}+\sqrt{x-1}\right)^2}{x^2-1}\times\frac{x^2-1}{2}-\sqrt{1-x^2}\)
\(A=\frac{x+1+2\sqrt{x^2-1}+x-1}{2}-\sqrt{1-x^2}\)
\(A=\frac{2x+2\sqrt{x^2-1}-2\sqrt{1-x^2}}{2}\)
\(A=\frac{2x+2\sqrt{x^2-1}+2\sqrt{x^2-1}}{2}\)
\(A=\frac{2x+4\sqrt{x^2-1}}{2}\)
\(A=x+2\sqrt{x^2-1}\)
\(P=\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}-\frac{\sqrt{x}+1}{\sqrt{x}-1}\right)\left(\frac{1}{2\sqrt{x}}-\frac{\sqrt{x}}{2}\right)^2\)
\(P=\left[-\frac{4\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right]\left(-\frac{\sqrt{x}}{2}+\frac{1}{2\sqrt{x}}\right)^2\)
\(P=\left[-\frac{4\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right]\left(\frac{1}{4x}+\frac{1}{4}-\frac{1}{2}\right)\)
\(P=-\frac{4\sqrt{x}.\left(\frac{1}{4x}-\frac{1}{2}+\frac{x}{4}\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(P=-\frac{4.\frac{x^2-2x+1}{4x}.\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}\)
\(P=-\frac{\frac{x^2-2x+1}{\sqrt{x}}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(P=-\frac{x^2-2x+1}{\sqrt{x}.\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(P=-\frac{\sqrt{x}.\left(x-1\right)}{x}\)
ĐK : x>0, x khác 1
\(A=\left(\frac{1}{\sqrt{x}+1}+\frac{2\left(1-\sqrt{x}\right)}{x\left(\sqrt{x}+1\right)-\left(\sqrt{x}+1\right)}\right):\left(\frac{\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}-\frac{2}{x-1}\right)\)
\(=\left(\frac{1}{\sqrt{x}+1}-\frac{2\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}\right):\left(\frac{1}{\sqrt{x}-1}-\frac{2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)
\(=\frac{\sqrt{x}+1-2}{\left(\sqrt{x}+1\right)^2}:\frac{\sqrt{x}+1-2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{\left(\sqrt{x}+1\right)^2}.\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\sqrt{x}-1}\)
\(=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)
