\(\sqrt{\frac{y+1}{\left(x+2\right)^2}}.\sqrt{\frac{\left(x-2\right)^6}{y^2-2y+1}}\)
\(=\frac{\sqrt{y+1}}{x+2}\sqrt{\frac{\left(x-2\right)^6}{\left(y-1\right)^2}}\)
\(=\frac{\sqrt{y+1}}{x+2}.\frac{\left(x-2\right)^6}{y-1}\)
\(=\frac{-\left(x-2\right)^5\sqrt{y+1}}{y-1}\)
1) \(\frac{3+2\sqrt{3}}{\sqrt{3}}+\frac{2+\sqrt{2}}{\sqrt{2}+1}-\left(\sqrt{2}+3\right)\)
2) \(0.1\sqrt{\left(-3\right)^2}\cdot\left[6\sqrt{\left(-\frac{1}{3}\right)^2}-\sqrt{\left(\sqrt{3}-2\right)^2}\right]^2\)
3) \(\left(\frac{3+2\sqrt{3}}{\sqrt{3}+2}+\frac{2+\sqrt{2}}{\sqrt{2}+1}\right)\div\left(1\div\frac{1}{\sqrt{2}+\sqrt{3}}\right)\)
4) \(\left(\frac{3\sqrt{2}+\sqrt{6}}{\sqrt{12}+2}-\frac{\sqrt{54}}{3}\right)\cdot\frac{2}{\sqrt{6}}\)
5) \(\sqrt{\frac{5+2\sqrt{6}}{5-2\sqrt{6}}}+\sqrt{\frac{5-2\sqrt{6}}{5+2\sqrt{6}}}\)
Đố: Cho \(\Delta ABC\), biết \(BC=a,AC=b,AB=c,\widehat{A}=\alpha,\widehat{B}=\beta,\widehat{C}=\gamma\) chứng minh:
a)\(\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}\) b) \(a^2=b^2+c^2-2bc\cos\alpha\)
c) \(\frac{a-b}{a+b}=\frac{\tan\left[\frac{1}{2}\left(\alpha-\beta\right)\right]}{\tan\left[\frac{1}{2}\left(\alpha+\beta\right)\right]}\)
d) Biết \(s=\frac{a+b+c}{2}\). Chứng minh \(\frac{\cot\frac{\alpha}{2}}{s-a}=\frac{\cot\frac{\beta}{2}}{s-b}=\frac{\cot\frac{\gamma}{2}}{s-c}\)
Rút gọn biểu thức
1) \(\frac{\sqrt{5+2\sqrt{6}}+\sqrt{8+2\sqrt{15}}}{\sqrt{7+2\sqrt{10}}}\)
2) \(\left(2+\frac{3+\sqrt{3}}{\sqrt{3}+1}\right)\left(2+\frac{3-\sqrt{3}}{\sqrt{3}-1}\right):\left(\sqrt{5}-2\right)\)
3) \(\left(\frac{15}{\sqrt{6}+1}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\right).\left(\sqrt{6}+11\right)\)
4) \(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{99}+\sqrt{100}}\)
5) \(\frac{1}{1-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}-...-\frac{1}{\sqrt{98}-\sqrt{99}}+\frac{1}{\sqrt{99}-\sqrt{100}}\)
6) \(\frac{1}{2+\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+...+\frac{1}{100\sqrt{99}+99\sqrt{100}}\)
7)\(\left(\sqrt{\frac{2}{3}}+\sqrt{\frac{3}{2}}+2\right)\left(\frac{\sqrt{2}+\sqrt{3}}{4\sqrt{2}}-\frac{\sqrt{3}}{\sqrt{2}+\sqrt{3}}\right)\left(24+8\sqrt{6}\right)\left(\frac{\sqrt{2}}{\sqrt{2}+\sqrt{3}}+\frac{\sqrt{3}}{\sqrt{2}-\sqrt{3}}\right)\)
các bạn giải chi tiết giúp mk nhé. Cảm ơn
1. a> Rút gọn biểu thức sau : A= \(5\left(\frac{1}{\sqrt{2-\sqrt{3}}}+\sqrt{3-\sqrt{5}}-\frac{\sqrt{10}}{2}\right)^2\)+ \(\left(\frac{1}{\sqrt{2+\sqrt{3}}}+\sqrt{3-\sqrt{5}}-\frac{\sqrt{6}}{2}\right)^2\)
b) Cho biểu thức B= \(\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}-\frac{\sqrt{x}-1}{\sqrt{x+1}}-\frac{8\sqrt{x}}{x-1}\right):\left(\frac{\sqrt{x}-x-3}{x-1}-\frac{1}{\sqrt{x}-1}\right)\)
Rút gọn biểu thức B và chứng minh B nhỏ hơn hoặc bằng 1 với mọi x lớn hơn hoặc bằng 0 và x khác 1
rút gọn:
a)\(\left(\frac{1}{2+2\sqrt{x}}+\frac{1}{2-2\sqrt{x}}-\frac{x^2+1}{1-x^2}\right)\times\left(1+\frac{1}{x}\right)\)
b)\(\left(\frac{2\sqrt{xy}}{x-y}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{x}+\sqrt{y}}\right)\times\frac{2\sqrt{x}}{\sqrt{x}+\sqrt{y}}+\frac{\sqrt{y}}{\sqrt{y}-\sqrt{x}}\)
c)\(\left(\frac{x-1}{\sqrt{x}-1}+\frac{x\sqrt{x}-1}{1-x}\right)\div\frac{\left(\sqrt{x}-1\right)^2+\sqrt{x}}{\sqrt{x}+1}\)
Rút gọn các biểu thức sau:
a)\(\frac{\sqrt{108x^3}}{\sqrt{12x}}\left(x>0\right)\)
b)\(\frac{\sqrt{13x^4y^6}}{\sqrt{208x^6y^6}}\left(x< 0;y\ne0\right)\)
c)\(\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}+\sqrt{y}\right)^2\)
d) \(\sqrt{\frac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}\left(x\ge\right)\)
e)\(\frac{x-1}{\sqrt{y}-1}.\sqrt{\frac{\left(y-2\sqrt{y}+1\right)^2}{\left(x-1\right)^4}}\left(y>0;x\ne1;y\ne1\right)\)
Cho \(x=\sqrt{6+2\sqrt{2}.\left(\sqrt{\frac{5}{2}-\sqrt{6}+\sqrt{\left(3\sqrt{a}+1\right)\left(2a-2\right)-\frac{6a^2+6\sqrt{a}-8a-4a\sqrt{a}}{\sqrt{a}-1}+8}}\right)}\) với a là số thực không âm
\(y=\frac{\frac{x-2}{x}+\frac{1}{x-2}}{12-8\sqrt{5}}.\left(-16\right)\)
So sánh x và y
Bài 1: Rút gọn
a. \(\left(5-2\sqrt{3}\right)^2+\left(5+2\sqrt{3}\right)^2\)
b. \(\left(\sqrt{5}+\sqrt{2}\right)^2-\left(2\sqrt{5}+1\right)\left(2\sqrt{5}-1\right)-\sqrt{40}\)
c. \(\left(\sqrt{2}-1\right)^2-\frac{2}{3}\sqrt{4}+\frac{4\sqrt{2}}{5}+\sqrt{1\frac{11}{15}}-\sqrt{2}\)
d. \(\left(\sqrt{6}-\sqrt{18}+5\sqrt{2}-\frac{1}{2}\sqrt{8}\right)2\sqrt{6}+2\sqrt{3}\)
e. \(\left(2\sqrt{3}-3\sqrt{2}\right)^2+6\sqrt{6}+3\sqrt{24}\)
Bài 2: Rút gọn
A =\(\left(\frac{1}{x-\sqrt{x}}+\frac{1}{\sqrt{x}-1}:\frac{\sqrt{x+1}}{x-2\sqrt{x}+1}\right)\)(x>0 ; x khác 1)
\(\hept{\begin{cases}4\sqrt{1+\frac{1}{x^2}+\frac{1}{\left(x+1\right)^2}}+5y=\left(\sqrt{y}+2\sqrt{y+1}\right)^2\\4\sqrt{1+\frac{1}{y^2}+\frac{1}{\left(y+1\right)^2}}+5x=\left(\sqrt{x}+2\sqrt{x+1}\right)^2\end{cases}}\)
