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

\(P=\left[-\frac{4\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right]\left(-\frac{\sqrt{x}}{2}+\frac{1}{2\sqrt{x}}\right)^2\)

\(P=\left[-\frac{4\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right]\left(\frac{1}{4x}+\frac{1}{4}-\frac{1}{2}\right)\)

\(P=-\frac{4\sqrt{x}.\left(\frac{1}{4x}-\frac{1}{2}+\frac{x}{4}\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)

\(P=-\frac{4.\frac{x^2-2x+1}{4x}.\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}\)

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

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

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

kết quả là \(\sqrt{x}\left(1-\sqrt{x}\right)\) phải k mọi người

ĐK : x>0, x khác 1

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

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

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

\(=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)

cho S=1-3+3²-3³+...+3⁹⁸-3⁹⁹                                                                                                                                       

a. Chứng minh: S chia hêt cho 20

b. Rút gọn S, từ đó suy ra 3¹⁰⁰ chia 4 dư 1

chịu

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

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

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

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

\(=\frac{x}{\sqrt{x}+1}\cdot\frac{x-1}{2}=\frac{x^2-x}{2\sqrt{x}+2}\)

vào câu hỏi tương tự sẽ có chi tiết hơn nha

Thắng Nguyễn  cảm ơn cách làm của bạn nhưng bạn chưa rút gọn hết

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

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

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

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

Nhưng dù sao cũng cảm ơn rất nhiều

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

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

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

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

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

Trả lời:

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

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

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

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

\(A=\left[\frac{x-1}{\sqrt{x}.\left(\sqrt{x}-1\right)}\right]\div\left[\frac{\sqrt{x}+1}{\left(\sqrt{x}-1\right).\left(\sqrt{x}+1\right)}\right]\)

\(A=\frac{x-1}{\sqrt{x}.\left(\sqrt{x}-1\right)}\div\frac{1}{\sqrt{x}-1}\)

\(A=\frac{x-1}{\sqrt{x}.\left(\sqrt{x}-1\right)}\times\frac{\sqrt{x}-1}{1}\)

\(A=\frac{x-1}{\sqrt{x}}\)

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