So sánh 456/461 và 123/128

a) Ta có : \(\dfrac{12}{47}>\dfrac{12}{53}>\dfrac{11}{53}\) \(\Leftrightarrow\dfrac{12}{47}>\dfrac{11}{53}\) b) Ta có : \(\dfrac{456}{461}=\dfrac{461-5}{461}=1-\dfrac{5}{461}\) \(\dfrac{123}{128}=\dfrac{128-5}{128}=1-\dfrac{5}{128}\) Do \(1-\dfrac{5}{461}>1-\dfrac{5}{128}\) \(\Rightarrow\dfrac{456}{461}>\dfrac{123}{128}\) c) Ta có: \(\dfrac{12}{47}\) > \(\dfrac{12}{48}=\dfrac{1}{4}\) \(\dfrac{19}{77}< \dfrac{19}{76}=\dfrac{1}{4}\) Do \(\dfrac{12}{47}>\dfrac{1}{4}>\dfrac{19}{77}\) \(\Rightarrow\dfrac{12}{47}>\dfrac{19}{77}\) d) Ta có : A=\(\dfrac{13^{15}+1}{13^{16}+1}\) \(\Leftrightarrow\) 13A=\(\dfrac{13.\left(13^{15}+1\right)}{13^{16}+1}\) \(\Leftrightarrow\) 13A=\(\dfrac{13^{16}+13}{13^{16}+1}\) \(=\dfrac{13^{16}+1+12}{13^{16}+1}=\dfrac{13^{16}+1}{13^{16}+1}+\dfrac{12}{13^{16}+1}\)
\(=1+\dfrac{12}{13^{16}+1}\) B=\(\dfrac{13^{16}+1}{13^{17}+1}\) \(\Leftrightarrow\) 13B=\(\dfrac{13.\left(13^{16}+1\right)}{13^{17}+1}\)

\(\Leftrightarrow\) 13B=\(\dfrac{13^{17}+13}{13^{17}+1}=\dfrac{13^{17}+1+12}{13^{17}+1}\) \(=\dfrac{13^{17}+1}{13^{17}+1}+\dfrac{12}{13^{17}+1}\) \(=1+\dfrac{12}{13^{17}+1}\)

Do \(1+\dfrac{12}{13^{16}+1}.>1+\dfrac{12}{13^{17}+1}\) nên 13A>13B \(\Rightarrow\) A>B