toan 9

M

mydream_1997

đừng ném đá nha:D:D:D:D
ta có [TEX]2012+2013<2012+2013+2\sqrt{2012.2013}[/TEX]
[TEX]\Leftrightarrow 2012+2013<(\sqrt{2012}+\sqrt{2013})^2[/TEX]
[TEX]\Leftrightarrow \sqrt{2012+2013}<\sqrt{2012}+\sqrt{2013}[/TEX] (ĐPCM)
 
1

12ab3csy

Is this thread?: Compare $\sqrt[]{2012+2013}$ and $\sqrt[]{2012}+\sqrt[]{2013}$
The answer: $(\sqrt[]{2012}+\sqrt[]{2013})^2$ = $2012+2013+2\sqrt[]{2012.2013}$ > $2012+2013$
So $\sqrt[]{2012}+\sqrt[]{2013}$ > $\sqrt[]{2012+2013}$
 
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N

nghgh97

Tổng quát: so sánh $\sqrt {A + B} $ và $\sqrt A + \sqrt B $ với A, B dương khác 0
Ta có:
${\left( {\sqrt {A + B} } \right)^2} = A + B$
${\left( {\sqrt A + \sqrt B } \right)^2} = A + B + 2\sqrt {AB} $
$ \Rightarrow {\left( {\sqrt {A + B} } \right)^2} < {\left( {\sqrt A + \sqrt B } \right)^2}$
$ \Leftrightarrow \sqrt {A + B} < \sqrt A + \sqrt B $
Áp dụng:
$\sqrt {2012 + 2013} < \sqrt {2012} + \sqrt {2013} $
 
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