Ta có:

\(A=\frac{13^{15}+1}{13^{16}+1}\Rightarrow13A=\frac{13^{16}+13}{13^{16}+1}=\frac{13^{16}+1+12}{13^{16}+1}=1+\frac{12}{13^{16}+1}\)

\(B=\frac{13^{16}+1}{13^{17}+1}\Rightarrow13B=\frac{13^{17}+13}{13^{17}+1}=\frac{13^{17}+1+12}{13^{17}+1}=1+\frac{12}{13^{17}+1}\)

Ta thấy:

\(13^{16}+1< 13^{17}+1\)

\(\Rightarrow\frac{12}{13^{16}+1}>\frac{12}{13^{17}+1}\)

\(\Rightarrow1+\frac{12}{13^{16}+1}>1+\frac{12}{13^{17}+1}\)

hay \(A>B\)

Vậy \(A>B.\)

Ta có: \(\frac{a}{b}< \frac{a+c}{b+c}\)

=> \(B=\frac{13^{16}+1}{13^{17}+1}< \frac{13^{16}+1+12}{13^{17}+1+12}=\frac{13^{16}+13}{13^{17}+13}=\frac{13\left(13^{15}+1\right)}{13\left(13^{16}+1\right)}=\frac{13^{15}+1}{13^{16}+1}=A\)

Vậy: \(A>B\) 

Ta có: \(13A=1+\frac{12}{13^{16}+1};13B=1+\frac{12}{13^{17}+1}\)

Do \(\frac{12}{13^{16}+1}>\frac{12}{13^{17}+1}\). Nên \(13A>13B\) 

Vậy \(A>B\)

B > A

Mk nghĩ thế thui

a) \(\dfrac{12}{47}\) và \(\dfrac{11}{53}\)

Ta có: \(\dfrac{11}{47}>\dfrac{11}{53}\) mà \(\dfrac{12}{47}>\dfrac{11}{47}\)

\(\Rightarrow\dfrac{12}{47}>\dfrac{11}{53}\)

a) Ta có :\(\dfrac{12}{47}>\dfrac{12}{48}=\dfrac{1}{4}=\dfrac{11}{44}>\dfrac{11}{53}\)

\(\Rightarrow\dfrac{12}{47}>\dfrac{11}{53}\)

b) Ta có : \(\dfrac{456}{461}=1-\dfrac{5}{461}\)

\(\dfrac{123}{128}=1-\dfrac{5}{128}\)

Vì \(\dfrac{5}{461}< \dfrac{5}{128}\Rightarrow1-\dfrac{5}{461}>1-\dfrac{5}{128}\)

\(\Rightarrow\dfrac{456}{461}>\dfrac{123}{128}\)

c) Ta có :\(\dfrac{12}{47}>\dfrac{12}{48}=\dfrac{1}{4}=\dfrac{19}{76}>\dfrac{19}{77}\)

=> \(\dfrac{12}{47}>\dfrac{19}{77}\)

d) Ta có : \(13A=13.\dfrac{13^{15}+1}{13^{16}+1}=\dfrac{13^{16}+13}{13^{16}+1}=\dfrac{13^{16}+1+12}{13^{16}+1}=1+\dfrac{12}{13^{16}+1}\)

\(13B=13.\dfrac{13^{16}+1}{13^{17}+1}=\dfrac{13^{17}+13}{13^{17}+1}=\dfrac{13^{17}+1+12}{13^{17}+1}=1+\dfrac{12}{13^{17}+1}\)

Ta thấy : \(\dfrac{12}{13^{16}+1}>\dfrac{12}{13^{17}+1}\Rightarrow1+\dfrac{12}{13^{16}+1}>1+\dfrac{12}{13^{17}+1}\Rightarrow\dfrac{13^{15}+1}{13^{16}+1}>\dfrac{13^{16}+1}{13^{17}+1}\)

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

a) \(\dfrac{17}{20}< \dfrac{18}{20}< \dfrac{18}{19}\Rightarrow\dfrac{17}{20}< \dfrac{18}{19}\)

b) \(\dfrac{19}{18}>\dfrac{19+2024}{18+2024}=\dfrac{2023}{2022}\Rightarrow\dfrac{19}{18}>\dfrac{2023}{2022}\)

c) \(\dfrac{135}{175}=\dfrac{27}{35}\)

\(\dfrac{13}{17}=\dfrac{26}{34}< \dfrac{26+1}{34+1}=\dfrac{27}{35}\)

\(\Rightarrow\dfrac{13}{17}< \dfrac{135}{175}\)

\(A=\dfrac{13^{15}+1}{13^{16}+1}\)

\(\Leftrightarrow13A=\dfrac{13^{16}+13}{13^{16}+1}=1+\dfrac{12}{13^{16}+1}\)

\(B=\dfrac{13^{16}+1}{13^{17}+1}\)

\(\Leftrightarrow13B=\dfrac{13^{17}+13}{13^{17}+1}=1+\dfrac{12}{13^{17}+1}\)

mà \(13^{16}+1< 13^{17}+1\)

nên A>B

Bài 1:

Ta có:

\(\left(\frac{1}{10}\right)^{15}=\left(\frac{1}{5}\right)^{3.5}=\left(\frac{1}{125}\right)^5\)

\(\left(\frac{3}{10}\right)^{20}=\left(\frac{3}{10}\right)^{4.5}=\left(\frac{81}{10000}\right)^5\)

Lại có:

\(\frac{1}{125}=\frac{80}{10000}< \frac{81}{10000}\Rightarrow\left(\frac{1}{125}\right)^5< \left(\frac{81}{10000}\right)^5\)

\(\Rightarrow\left(\frac{1}{10}\right)^{15}< \left(\frac{3}{10}\right)^{20}\)

Bài 2:

Ta có:

\(A=\frac{13^{15}+1}{13^{16}+1}\Rightarrow13A=\frac{13^{16}+13}{13^{16}+1}=1+\frac{12}{13^{16}+1}\)

\(B=\frac{13^{16}+1}{13^{17}+1}\Rightarrow13B=\frac{13^{17}+13}{13^{17}+1}=1+\frac{12}{13^{17}+1}\)

Mà \(\frac{12}{13^{16}+1}>\frac{12}{13^{17}+1}\)

\(\Rightarrow1+\frac{12}{13^{16}+1}>1+\frac{12}{13^{17}+1}\)

\(\Rightarrow13A>13B\Rightarrow A>B\)