\(=\frac{\sqrt{3}+\sqrt{2}-1}{2+\sqrt{6}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+1}\left(\frac{2\sqrt{3}+\sqrt{18}+2\sqrt{3}-\sqrt{18}}{4-6}\right)-\frac{1}{\sqrt{2}}.\)

\(=\frac{\sqrt{3}+\sqrt{2}-1}{2+\sqrt{6}}-\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+1}.\left(2\sqrt{3}\right)-\frac{1}{\sqrt{2}}\)

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

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

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

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

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

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

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

\(\sqrt{2}A=\sqrt{4+2\sqrt{3}}-\sqrt{12-6\sqrt{3}}+\frac{2}{\sqrt{3}+1}\)

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

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

\(A=\sqrt{6}-\sqrt{2}+\frac{\sqrt{2}}{1+\sqrt{3}}=\frac{3\sqrt{6}-3\sqrt{2}}{2}\)

*****~~~~~~~~~~*****

 \(\frac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}+\frac{6+\sqrt{6}}{\sqrt{6}+1}\)

\(=\frac{\sqrt{3}\left(\sqrt{5}-2\right)}{\sqrt{5}-2}+\frac{\sqrt{6}\left(\sqrt{6}+1\right)}{\sqrt{6}+1}\)

\(=\sqrt{3}+\sqrt{6}\)

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

*****~~~~~~~~~~*****

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

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

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

(Chúc bạn học tốt nha!)

1.052631148

có hiểu rút gọn là j ko thế

\(=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{\frac{\sqrt{2}\left(\sqrt{\sqrt{3}}+1\right)}{\sqrt{2}\left(\sqrt{\sqrt{3}}-1\right)}}=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{\frac{\left(\sqrt{\sqrt{3}}+1\right)^2}{\left(\sqrt{\sqrt{3}}-1\right)\left(\sqrt{\sqrt{3}}+1\right)}}\)\(=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{\frac{\sqrt{3}+1+2\sqrt{\sqrt{3}}}{\sqrt{3}-1}}\)\(=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{\frac{\left(\sqrt{3}+1+2\sqrt{\sqrt{3}}\right)\left(\sqrt{3}+1\right)}{2}}=\frac{\sqrt{5}\left(\sqrt{6}+1\right)}{2+\sqrt{3}+\sqrt{\sqrt{3}}+\sqrt{3\sqrt{3}}}\)

\(=\frac{\sqrt{30}+\sqrt{5}}{\left(\sqrt{3}+1\right)\left(\sqrt{\sqrt{3}}+1\right)+1}=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{\sqrt{3}}+1\right)\left(\sqrt{\sqrt{3}}-1\right)+\sqrt{\sqrt{3}}-1}\)

\(=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)+\sqrt{\sqrt{3}}-1}\)

\(=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)\left(\sqrt{\sqrt{3}}-1\right)}{\left(\sqrt{\sqrt{3}}+1\right)\left(\sqrt{\sqrt{3}}-1\right)}\)

\(=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)^2\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\)

\(=\frac{\left(\sqrt{30}+\sqrt{5}\right)\left(\sqrt{\sqrt{3}}-1\right)^2\left(\sqrt{3}+1\right)}{2}\)\(=2\sqrt{30}+2\sqrt{5}+\sqrt{90}+\sqrt{15}-\sqrt{90\sqrt{3}}-\sqrt{30\sqrt{3}}-\sqrt{15\sqrt{3}}-\sqrt{5\sqrt{3}}\)

mởi tay ùi,có gì thiếu tự giải tiếp ^^

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 \(1-\sqrt{9}\)

ĐK \(x\ne\left\{2;3\right\}\)

Ta có \(A=\frac{\sqrt{x}+2}{\sqrt{x}-3}-\frac{\sqrt{x}+1}{\sqrt{x}-2}-3.\frac{\sqrt{x}-1}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\)

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

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