a: \(3^4=3^4;9^3=\left(3^2\right)^3=3^{2\cdot3}=3^6\)

mà \(3^4<3^6\)

nên \(3^4<9^3\)

b: \(A=1+2+2^2+\cdots+2^{2017}\)

=>\(2A=2+2^2+2^3+\cdots+2^{2018}\)

=>\(2A-A=2+2^2+2^3+\cdots+2^{2018}-1-2-2^2-\cdots-2^{2017}\)

=>\(A=2^{2018}-1\)

=>A=B

c: \(16^{19}=\left(2^4\right)^{19}=2^{4\cdot19}=2^{76};8^{25}=\left(2^3\right)^{25}=2^{3\cdot25}=2^{75}\)

mà \(2^{76}<2^{75}\)

nên \(16^{19}<8^{25}\)

d: \(5^{23}=5\cdot5^{22}<6\cdot5^{22}\)

e: \(5^{36}=\left(5^3\right)^{12}=125^{12}\)

\(11^{24}=\left(11^2\right)^{12}=121^{12}\)

mà 125>121

nên \(5^{36}>11^{24}\)

Ta có: \(A=\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

\(=\left(2^8-1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

\(=\left(2^{16}-1\right)\left(2^{16}+1\right)\)

\(=2^{32}-1< 2^{32}\)

\(\Leftrightarrow A< B\)

A=(3-1)(3+1)(3²+1)(3⁴+1)(3⁸+1)(3¹⁶+1)

A= (3²-1)(3²+1)(3⁴+1)(3⁸+1)(3¹⁶+1)

A= (3⁴-1)(3⁴+1)(3⁸+1)(3¹⁶+1)

A= (3^8_1)(3⁸+1)(3¹⁶+1)

A= (3¹⁶-1)(3¹⁶+1)

A=3²²-1

Vậy A=B

A = (2 - 1)(2 + 1)(2^2 + 1 )(2^4 + 1 ) (2^8 + 1)(2^16 + 1)  ( nhân vói 2 - 1 = 1 Gía không thay dổi)

A = ( 2 ^2 - 1 )(2^2 + 1 )(2^4  + 1 )(2^8 + 1 )(2^16 + 1 )

A = ( 2^4 - 1 )(2^4 + 1)(2^8 + 1)(2^16 + 1)

A = (2^8 - 1)(2^8 + 1)(2^16 + 1)

A = (2^16 - 1)(2^16 + 1 )

A = 2^32 - 1 <2^32 = B 

VẬy A < B

A<B

\(A=\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\)

\(A=\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\)

\(A=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\)

\(A=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right)\)

\(A=\left(2^8-1\right)\left(2^8+1\right)\)

\(A=2^{16}-1< 2^{16}\)

Mình làm theo cách tính nhé !

\(A=\left(2+1\right).\left(2^2+1\right).\left(2^4+1\right).\left(2^8+1\right)\)

\(A=3.\left(4+1\right).\left(16+1\right).\left(256+1\right)\)

\(A=3.5.17.257\)

\(\Rightarrow A=65535\) 

\(B=2^{16}=65536\) 

Từ đó \(\Rightarrow A< B\)

A = 3(2^2+1)(2^4+1)(2^8+1)(2^16+1)+1 

 =(2²-1)(2²+1)(2⁴+1)(2⁸+1)(2¹⁶+1)+1

=(2⁴-1)(2⁴+1)(2⁸+1)(2¹⁶+1)+1

=(2⁸-1)(2⁸+1)(2¹⁶+1)+1

=(2¹⁶-1)(2¹⁶+1)+1

=2³²-1+1

=2³² = B

vậy A=B