\(\sqrt{1^3+2^3}=1+2\)
\(\Leftrightarrow\sqrt{1+8}=3\)
\(\Leftrightarrow\sqrt{9}=3\)
mà \(\sqrt{9}=\sqrt{3^2}=\left|3\right|=3\)
\(\Leftrightarrow3=3\)
\(\Rightarrow\sqrt{1^3+2^3}=1+2\)
mấy bài khác chị giải tương tự là ra.
Chứng minh rằng \(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+...+\frac{1}{100\sqrt{99}+99\sqrt{100}}< 1\)
1.Chứng minh: \(\frac{1}{2\cdot\sqrt{1}}+\frac{1}{3\cdot\sqrt{2}}+\frac{1}{4\cdot\sqrt{3}}+...+\frac{1}{2012\cdot\sqrt{2011}}+\frac{1}{2013\cdot\sqrt{2012}}\)\(< 2\)
2.Chứng minh: A= \(\frac{1}{3\cdot\left(\sqrt{1}+\sqrt{2}\right)}+\frac{1}{5\cdot\left(\sqrt{2}+\sqrt{3}\right)}+...+\frac{1}{97\cdot\left(\sqrt{48}+\sqrt{49}\right)}\)\(< \frac{1}{2}\)
thực hiện phép tính:
1, \(\dfrac{\sqrt{15}-\sqrt{5}}{\sqrt{3}-1}-\dfrac{5-2\sqrt{5}}{2\sqrt{5}-4}\)
2,\(\dfrac{\sqrt{3}+1}{\sqrt{3}-1}+\dfrac{\sqrt{3}-1}{\sqrt{3}+1}\)
3,\(\dfrac{2\sqrt{8}-\sqrt{12}}{\sqrt{18}-\sqrt{48}}+\dfrac{\sqrt{5}+\sqrt{27}}{\sqrt{30}-\sqrt{2}}\)
4,\(\dfrac{3-\sqrt{3}}{2\sqrt{3}-1}+\dfrac{3+\sqrt{3}}{2\sqrt{3}-1}\)
5,\(\dfrac{2\sqrt{3}-4}{\sqrt{3}-1}+\dfrac{2\sqrt{2}-1}{\sqrt{2}-1}-\dfrac{1+\sqrt{6}}{\sqrt{2}+\sqrt{3}}\)
chứng minh
a) \(\sqrt{\frac{2-\sqrt{3}}{2+\sqrt{3}}}\) +\(\sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}}\) =4
b) (1+\(\frac{a+\sqrt{a}}{\sqrt{a}+1}\) ).(1-\(\frac{a-\sqrt{a}}{\sqrt{a}-1}\))=1-a
Chứng minh :
\(\sqrt{1^3+2^3}=1+2\)
\(\sqrt{1^3+2^3+3^3}=1+2+3\)
\(\sqrt{1^3+2^3+3^3+4^3}=1+2+3+4\)
Viết tiếp một số đẳng thức tương tự ?
Tính tổng sau: \(S=\dfrac{1}{2+\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{4}}+...+\dfrac{1}{100\sqrt{99}+99\sqrt{100}}\)
Tính giá trị của P = \(\left(\dfrac{\sqrt{x-1}}{3+\sqrt{x-1}}+\dfrac{x+8}{10-x}\right):\left(\dfrac{3\sqrt{x-1}+1}{x-3\sqrt{x-1}-1}-\dfrac{1}{\sqrt{x-1}}\right)\)khi x=\(\sqrt[4]{\dfrac{3+2\sqrt{2}}{3-2\sqrt{2}}}-\sqrt[4]{\dfrac{3-2\sqrt{2}}{3+2\sqrt{2}}}\)
\(\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}\)
\(\dfrac{1}{\sqrt{3}-1}-\dfrac{1}{\sqrt{3}+1}\)
\(2\sqrt{5}-3\sqrt{45}+\sqrt{500}\)
\(\dfrac{1}{\sqrt{3}+\sqrt{2}}-\dfrac{\sqrt{15}-\sqrt{12}}{\sqrt{5}-2}\)
\(\dfrac{1}{2+\sqrt{3}}-\dfrac{1}{2-\sqrt{3}}+5\sqrt{3}\)
\(\sqrt{3}-\sqrt{4+2\sqrt{3}}\)
\(\dfrac{5-\sqrt{5}}{\sqrt{5}-1}-\dfrac{4}{\sqrt{5}+1}\)
\(\sqrt{37-20\sqrt{3}+\sqrt{37+20\sqrt{3}}}\)
a)cho a>b>0 chứng minh rằng : \(\dfrac{1}{a+b}\le\dfrac{1}{2\sqrt{ab}}\)
b) Chứng minh \(\dfrac{\sqrt{2}-\sqrt{1}}{3}+\dfrac{\sqrt{3}-\sqrt{2}}{5}+\dfrac{\sqrt{4}-\sqrt{3}}{7}+...+\dfrac{\sqrt{2011}-\sqrt{2010}}{4021}< \dfrac{1}{2}\)
giúp mk vs
