So sánh:

\(A=-\frac{9}{10^{2012}}-\frac{19}{10^{2011}}\) và \(B=-\frac{9}{10^{2011}}-\frac{19}{10^{2012}}\)

Ta có: 

\(A=-\frac{9}{10^{2012}}-\frac{19}{10^{2011}}=-\frac{1}{10^{2011}}\left(\frac{9}{10}+19\right)=-\frac{1}{10^{2011}}.\frac{199}{10}\)

\(B=-\frac{9}{10^{2011}}-\frac{19}{10^{2012}}=-\frac{1}{10^{2011}}\left(9+\frac{19}{10}\right)=-\frac{1}{10^{2011}}.\frac{109}{10}\)

Vì \(\frac{199}{10}>\frac{109}{10}\Rightarrow\frac{1}{10^{2011}}.\frac{199}{10}>\frac{1}{10^{2011}}.\frac{109}{10}\Rightarrow-\frac{1}{10^{2011}}.\frac{199}{10}< -\frac{1}{10^{2011}}.\frac{109}{10}\)

Vậy nên A<B

A < B nha!

Có : \(A=\frac{10^{2012}-10}{10^{2013}-10}\)

\(\Leftrightarrow10A=\frac{10^{2013}-100}{10^{2013}-10}\)

\(\Leftrightarrow10A=\frac{10^{2013}-10-90}{10^{2013}-10}\)

\(\Leftrightarrow10A=1-\frac{90}{10^{2013}-10}\)

Có : \(B=\frac{10^{2011}+10}{10^{2012}+10}\)

\(\Leftrightarrow10B=\frac{10^{2012}+100}{10^{2012}+10}\)

\(\Leftrightarrow10B=\frac{10^{2012}+10+90}{10^{2012}+10}\)

\(\Leftrightarrow B=1+\frac{90}{10^{2012}+10}\)

Ta thấy : \(1-\frac{90}{10^{2013}-10}< 1\)

              \(1+\frac{90}{10^{2012}+10}>1\)

\(\Leftrightarrow1-\frac{90}{10^{2013}-10}< 1+\frac{90}{10^{2012}+10}\)

\(\Leftrightarrow A< B\)

\(B< \frac{10^{2012}+1+9}{10^{2013}+1+9}=\frac{10^{2012}+10}{10^{2013}+10}=\frac{10\left(10^{2011}+1\right)}{10\left(10^{2012}+1\right)}=\frac{10^{2011}+1}{10^{2012}+1}=A\)

Vậy A > B

Áp dụng bất đẳng thức :

\(\frac{a}{b}< 1\Leftrightarrow\frac{a}{b}< \frac{a+m}{b+m}\)

Ta có :

\(B=\frac{10^{2012}+1}{10^{2013}+1}< \frac{10^{2012}+1+9}{10^{2013}+1+9}=\frac{10^{2012}+10}{10^{2013}+10}=\frac{10\left(10^{2011}+1\right)}{10\left(10^{2012}+1\right)}=\frac{10^{2011}+1}{10^{2012}+1}=A\)

\(\Leftrightarrow B< A\)

cảm ơn nhiều

Cho C=\(10^{2010}+\frac{1}{10^{2010}}\)

Xét \(A_1=10^{2010}+\frac{1}{10^{2011}}\)và \(B^{ }_1=10^{2011}+\frac{1}{10^{2012}}\)

Ta có \(A_1-C=10^{2010}+\frac{1}{10^{2010}}-10^{2010}-\frac{1}{10^{2010}}\)

         \(A_1-C=10.\left(\frac{1}{10^{2011}}-\frac{1}{10^{2010}}\right)\)

Giair tượng tự ta được \(B_1-C=10^{2010}.\left(9+\frac{1}{10^{2012}}-\frac{1}{10^{2010}}\right)\)

Ta thấy \(\frac{1}{10^{2012}}-\frac{1}{10^{2010}}

Ta có : \(M=-\dfrac{7}{10^{2011}}+\dfrac{-15}{10^{2012}}\) và \(N=\dfrac{-15}{10^{2011}}+\dfrac{-8}{10^{2012}}\)

Xét \(M=-\dfrac{7}{10^{2011}}-\dfrac{15}{10^{2012}}=-\dfrac{1}{10^{2011}}\left(7+\dfrac{15}{10}\right)=-\dfrac{1}{10^{2011}}\cdot\dfrac{17}{2}\).

Xét \(N=-\dfrac{15}{10^{2011}}-\dfrac{8}{10^{2012}}=-\dfrac{1}{10^{2011}}\left(15+\dfrac{8}{10}\right)=-\dfrac{1}{10^{2011}}\cdot\dfrac{79}{5}\).

Ta cũng có : \(\dfrac{M}{N}=\dfrac{-\dfrac{1}{10^{2011}}\cdot\dfrac{17}{2}}{-\dfrac{1}{10^{2011}}\cdot\dfrac{79}{5}}=\dfrac{\dfrac{17}{2}}{\dfrac{79}{5}}=\dfrac{85}{158}\)

\(\Rightarrow M=\dfrac{85}{158}N\). Mà \(\dfrac{85}{158}< 1\) nên \(M< N\).

Vậy : \(M< N\).

Có \(\hept{\begin{cases}A=\frac{-9}{10^{2012}}+\frac{-19}{10^{2011}}\\B=\frac{-19}{10^{2012}}+\frac{-9}{10^{2011}}\end{cases}}\)

\(\Rightarrow\)A-B=\(\frac{10}{10^{2011}}-\frac{10}{10^{2012}}=\frac{1}{10^{2010}}-\frac{1}{10^{2011}}>0\)

\(\Rightarrow A>B\)