Bài 1:

a, \(4\sqrt{3+2\sqrt{2}}-\sqrt{57+40\sqrt{2}}\)

\(=4\sqrt{\left(\sqrt{2}+1\right)^2}-\sqrt{\left(4\sqrt{2}+5\right)^2}\)

\(=4\left(\sqrt{2}+1\right)-4\sqrt{2}-5\)

\(=4\sqrt{2}+4-4\sqrt{2}-5=-1\)

b, \(B=\sqrt{1100}-7\sqrt{44}+2\sqrt{176}-\sqrt{1331}\)

\(=10\sqrt{11}-14\sqrt{11}+8\sqrt{11}-11\sqrt{11}=-7\sqrt{11}\)

c, \(C=\sqrt{\left(1-\sqrt{2002}\right)^2}.\sqrt{2003+2\sqrt{2002}}\)

\(=\left(1-\sqrt{2002}\right).\sqrt{\left(\sqrt{2002}+1\right)^2}\)

\(=\left(1-\sqrt{2002}\right).\left(\sqrt{2002}+1\right)=-2001\)

Câu d bạn kiểm tra lại đề bài nhé.

Bài 2:

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

a, ĐK: \(x\ge0,x\ne1\)

b, ĐK: \(x\ge0,x\ne1\)

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

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

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

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

\(=\frac{4-4\sqrt{x}}{4\left(x-1\right)}=\frac{4\left(1-\sqrt{x}\right)}{4\left(1-x\right)}=\frac{1-\sqrt{x}}{1-x}\)

Thay \(x=3\left(TM\right)\)vào A ta có: \(A=\frac{1-\sqrt{3}}{3-1}=\frac{1-\sqrt{3}}{2}\)

Vậy với \(x=3\)thì \(A=\frac{1-\sqrt{3}}{2}\)

c, \(\left|A\right|=\frac{1}{2}\Leftrightarrow\orbr{\begin{cases}A=\frac{1}{2}\\A=-\frac{1}{2}\end{cases}}\)

TH1: \(A=\frac{1}{2}\)\(\Leftrightarrow\frac{1-\sqrt{x}}{x-1}=\frac{1}{2}\Leftrightarrow2-2\sqrt{x}=x-1\)\(\Leftrightarrow x-1-2+2\sqrt{x}=0\)\(\Leftrightarrow x+2\sqrt{x}-3=0\Leftrightarrow\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)=0\)\(\Leftrightarrow\orbr{\begin{cases}\sqrt{x}-1=0\\\sqrt{x}+3=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x=1\left(TM\right)\\\sqrt{x}=-3\left(L\right)\end{cases}}}\)

TH2: \(A=-\frac{1}{2}\Leftrightarrow\frac{1-\sqrt{x}}{x-1}=-\frac{1}{2}\)\(\Leftrightarrow2-2\sqrt{x}=1-x\Leftrightarrow-x+1-2+2\sqrt{x}=0\)\(\Leftrightarrow-x-1+2\sqrt{x}=0\Leftrightarrow x-2\sqrt{x}+1=0\)\(\Leftrightarrow\left(\sqrt{x}+1\right)^2=0\Leftrightarrow\sqrt{x}=-1\left(L\right)\)

Vậy với \(x=1\)thì \(\left|A\right|=\frac{1}{2}\)

Cám ơn bạn nhiều nha!!!

\(\sqrt{1+\frac{1}{\left(n-1\right)^2}+\frac{1}{n^2}}=1-\frac{1}{n}+\frac{1}{n-1}\) dựa vào mà làm

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

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

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

Áp dụng đẳng thức vừa chứng minh vào phương trình trên ta được:

\(1-\frac{1}{3}+\frac{1}{2}+1-\frac{1}{4}+\frac{1}{3}+...+1-\frac{1}{n}+\frac{1}{n-1}=2001\frac{2001}{4006}\)

<=>(1+1+1+...+1(n-2 số 1))\(+\frac{1}{2}-\frac{1}{n}=2001\frac{2001}{4006}\)

<=>\(n-2+\frac{1}{2}-\frac{1}{n}=2001\frac{2001}{4006}\)

=>4006n.(n-2)+2003n-4006=8018007

=>4006n²-8012n+2003n-4006=8018007

=>4006n²-6009n-8022013=0

@@số to thế

đẹp trai thì cũng đi tù thôi em ạ

đẹp trai--> đi tù

Logic lạ đó :))

1. Trục căn thức ở mẫu:

\(A=\frac{1}{1+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{9}}+\frac{1}{\sqrt{9}+\sqrt{13}}+....+\frac{1}{\sqrt{2001}+\sqrt{2005}}+\frac{1}{\sqrt{2005}+\sqrt{2009}}\)

=\(\frac{\sqrt{5}-1}{4}+\frac{\sqrt{9}-\sqrt{5}}{4}+\frac{\sqrt{13}-\sqrt{9}}{4}+....+\frac{\sqrt{2005}-\sqrt{2001}}{4}+\frac{\sqrt{2009}-\sqrt{2005}}{4}\)

\(=\frac{\sqrt{2009}-1}{4}\)

2/ \(x=\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\)

=> \(x^3=\left(\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\right)^3\)

\(=3+2\sqrt{2}+3-2\sqrt{2}+3\left(\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\right).\sqrt[3]{3+2\sqrt{2}}.\sqrt[3]{3-2\sqrt{2}}\)

\(=6+3x\)

=> \(x^3-3x=6\)

=> \(B=x^3-3x+2000=6+2000=2006\)

\(A=\frac{1-\sqrt{5}}{1-5}+\frac{\sqrt{5}-\sqrt{9}}{5-9}+\frac{\sqrt{9}-\sqrt{13}}{9-13}+...+\frac{\sqrt{2001}-\sqrt{2005}}{2001-2005}\)

\(A=\frac{1-\sqrt{5}+\sqrt{5}-\sqrt{9}+\sqrt{9}-\sqrt{13}+...+\sqrt{2001}-\sqrt{2005}}{-4}\)

\(A=\frac{1-\sqrt{2005}}{-4}=\frac{\sqrt{2005}-1}{4}\)