`6\sqrt(2/3)-\sqrt(24)+2\sqrt(3/8)+2\sqrt(1/6)`

`=6. \sqrt6/3 - \sqrt(2^2 .6) + 2. \sqrt(24)/8 + 2. \sqrt6/6`

`=2\sqrt6-2\sqrt6+ \sqrt6/2 + \sqrt6/3`

`=\sqrt6/2+\sqrt6/3`

`=(3\sqrt6+2\sqrt6)/6`

`=(5\sqrt6)/6`

Ta có: \(6\sqrt{\dfrac{2}{3}}-\sqrt{24}+2\sqrt{\dfrac{3}{8}}+2\sqrt{\dfrac{1}{6}}\)

\(=\dfrac{6\sqrt{2}}{\sqrt{3}}-2\sqrt{6}+2\cdot\dfrac{\sqrt{3}}{2\sqrt{2}}+\dfrac{2}{\sqrt{6}}\)

\(=2\sqrt{6}-2\sqrt{6}+\dfrac{\sqrt{3}}{\sqrt{2}}+\dfrac{\sqrt{2}}{\sqrt{3}}\)

\(=\dfrac{\sqrt{6}}{2}+\dfrac{\sqrt{6}}{3}=\dfrac{5\sqrt{6}}{6}\)

\(\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+\sqrt{16}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+2\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(=\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)+\left(\sqrt{4}+\sqrt{6}+\sqrt{8}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=1+\frac{\sqrt{2}\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=1+\sqrt{2}\)

với n >0, ta có :

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

Gọi biểu thức đã cho là A

\(A=\frac{1}{-\left(\sqrt{2}-\sqrt{1}\right)}-\frac{1}{-\left(\sqrt{3}-\sqrt{2}\right)}+...+\frac{1}{-\left(\sqrt{8}-\sqrt{7}\right)}-\frac{1}{-\left(\sqrt{9}-\sqrt{8}\right)}\)

\(A=-\frac{1}{\sqrt{2}-\sqrt{1}}+\frac{1}{\sqrt{3}-\sqrt{2}}-...-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{9}-\sqrt{8}}\)

\(A=-\left(\sqrt{2}+\sqrt{1}\right)+\left(\sqrt{3}+\sqrt{2}\right)-...-\left(\sqrt{8}+\sqrt{7}\right)+\left(\sqrt{9}+\sqrt{8}\right)\)

\(A=-\sqrt{1}+\sqrt{9}=2\)

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

\(b,\frac{2+\sqrt{3}}{1-\sqrt{4-2\sqrt{3}}}+\frac{2-\sqrt{3}}{1+\sqrt{4+2\sqrt{3}}}\)

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

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

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

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

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

\(=\frac{4+4\sqrt{3}+3+4-4\sqrt{3}+3}{4-3}\)

\(=14\)

\(a,\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+\sqrt{16}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(=\frac{\sqrt{2}+\sqrt{3}+4+\sqrt{6}+\sqrt{8}}{\sqrt{2}+\sqrt{3}+2}\)

\(=\frac{\sqrt{2}+\sqrt{3}+2}{\sqrt{2}+\sqrt{3}+2}+\frac{\sqrt{2}.\sqrt{2}+\sqrt{2}.\sqrt{3}+\sqrt{2}.2}{\sqrt{2}+\sqrt{3}+2}\)

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

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

\(12\sqrt{\frac{4}{3}}-\frac{8+2\sqrt{2}}{3-\sqrt{2}}-\frac{4-6\sqrt{2}}{\sqrt{2}}+\frac{\sqrt{3}}{\sqrt{3}-2}\)

\(=12.\frac{2}{\sqrt{3}}-\frac{\left(3+\sqrt{2}\right)\left(8-2\sqrt{2}\right)}{9-2}-\frac{\sqrt{2}\left(4-6\sqrt{2}\right)}{2}+\frac{\sqrt{3}\left(\sqrt{3}-2\right)}{3-4}\)

\(=8\sqrt{3}-\left(4+2\sqrt{2}\right)-\left(2\sqrt{2}-6\right)+\left(-3-2\sqrt{3}\right)\)

\(=8\sqrt{3}-4-2\sqrt{2}-2\sqrt{2}+6-3-2\sqrt{3}\)

\(=6\sqrt{3}-4\sqrt{2}-1\)

ko bit

=1.414213562

căn 2 (((: Hướng Dẫn: Tự nghĩ :v

Phân tích mỗi hạng tử theo kiểu như dưới đây

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

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

Khi đó mọi mẫu đều bằng -1

Bạn tiếp tục làm và kết quả nhận được là \(1-\sqrt{9}\)

a) \(\frac{\sqrt{6}+\sqrt{14}}{2\sqrt{3}+\sqrt{28}}=\frac{\sqrt{6}+\sqrt{14}}{\sqrt{2}\left(\sqrt{6}+\sqrt{14}\right)}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\)

b) \(\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+\sqrt{16}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)+\sqrt{2}\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)

\(\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=\sqrt{2}+1\)