\(2^{91}=\left(2^{13}\right)^7=73728^7\)

\(5^{35}=\left(5^5\right)^7=3125^7\) nhỏ hơn \(73728^7\)

\(\Rightarrow2^{91}\) lớn hơn \(5^{35}\)

\(b,3^{400}=\left(3^4\right)^{100}=81^{100}\\ 4^{300}=\left(4^3\right)^{100}=64^{100}\\ Vì:81^{100}>64^{100}\left(Do:81>64\right)\\ \Rightarrow3^{400}>4^{300}\)

kết bạn với nhau được không dương

a, 24^10 < 3^30 + 4^30 + 5^30

b, 2^100 < 5^50 < 3^75.

Từ đầu bài 

=> 5²S=5²+5⁴+5⁶+...+5²⁰²

=>5²S-S= (5²+5⁴+5⁶+...+5²⁰²)-(1+5²+5⁴+...+5²⁰⁰)

=>  24.S = 5²⁰²-1

=>     S  = \(\frac{5^{202}-1}{24}\)

Ta có : 

\(S=\frac{3}{2}+\frac{4}{3}+\frac{5}{4}+\frac{6}{5}+\frac{7}{6}+\frac{8}{7}+\frac{9}{8}+\frac{10}{9}+\frac{11}{10}+\frac{12}{11}\)

\(S=\frac{2+1}{2}+\frac{3+1}{3}+\frac{4+1}{4}+...+\frac{11+1}{11}\)

\(S=\left(1+\frac{1}{2}\right)+\left(1+\frac{1}{3}\right)+\left(1+\frac{1}{4}\right)+...+\left(1+\frac{1}{11}\right)\)

\(S=\left(1+1+1+...+1\right)+\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{11}\right)\)

\(S=10+\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{11}\right)>10\) 

\(\Rightarrow\)\(S>10\) 

Vậy \(S>10\)

Chúc bạn học tốt ~ 

Chắc mình phải lấy giấy vệ sinh thắt cổ tự tủ mất

\(A=\dfrac{5^{10}+1}{5^{11}+1}\)

=>\(5\cdot A=\dfrac{5^{11}+5}{5^{11}+1}=\dfrac{5^{11}+1+4}{5^{11}+1}=1+\dfrac{4}{5^{11}+1}\)

\(B=\dfrac{5^9+1}{5^{10}+1}\)

=>\(5B=\dfrac{5^{10}+5}{5^{10}+1}=1+\dfrac{4}{5^{10}+1}\)

\(5^{11}+1>5^{10}+1\)

=>\(\dfrac{4}{5^{11}+1}< \dfrac{4}{5^{10}+1}\)

=>\(\dfrac{4}{5^{11}+1}+1< \dfrac{4}{5^{10}+1}+1\)

=>5A<5B

=>A<B