Ta có: 31^11 < 32^11 và 17^14 > 16^14
=> 32^11=(2^5)^11=2^55
=>16^14= (2^4)^14=2^56
Ta thấy : 55^56
=>2^55 < 2^56
=> 32^11 < 16^14
Tức : 31^11 < 17^14
Chúc bạn học tốt!

\(32^{11}=\left(2^5\right)^{11}=2^{55}\\ 16^{14}=\left(2^4\right)^{14}=2^{56}\\ Ta.có:2^{55}< 2^{56}\Rightarrow32^{11}< 16^{14}\\ Mà:31^{11}< 32^{11};16^{14}< 17^{14}\Rightarrow31^{11}< 17^{14}\)

\(8^5=\left(2^3\right)^5=2^{15}=2.2^{14}\)

\(3.4^7=3.\left(2^2\right)^7=3.2^{14}\)

Vì 2 < 3 nên 8⁵ < 3 . 4⁷

8^5<3.4^7

Câu 1: 3^23  >    5^12

Câu 2: 3^36 < 2^8.11^4

4⁵⁰=  ( 4³ ) ^50/3 = 64 ^50/3

8³⁰ =( 8² ) ¹⁵ = 64¹⁵

ta có 50/3 > 15 => 4⁵⁰ > 8³⁰

\(4^{50}\)= \(\left(2^2\right)^{^{50}^{ }}\)\(=2^{100}\)

\(8^{30}=\left(2^3\right)^{30}=2^{90}\)

vì \(2^{100}>2^{90}\)nên\(4^{50}>8^{30}\)

\(333^{444}=333^{3^{111}}\) 

\(444^{333}=444^{3^{111}}\)

Vì \(444^{3^{111}}>333^{3^{111}}\)

=> \(333^{444}< 444^{333}\)

Ta có: \(333^{444}=\left(333^4\right)^{111}\)

           \(444^{333}=\left(444^3\right)^{111}\)

Vì 333⁴⁴⁴ và 444³³³ có cùng số mũ là 111. nên ta so sánh 333⁴ và 444³

333⁴=(3.111)⁴=3⁴.111⁴=81.111⁴

444³=(4.111)³=4³.111³=64.111³

Vì 81.111⁴>64.111³ => 333⁴>444³

=> 333⁴⁴⁴ > 444³³³

\(^{333^{444}>444^{333}}\)

3³⁰⁰ > 2³⁰⁰ ( vì 3 > 2 ).

3^200 = (3^2)^100 = 9^100 

2^300 = (2^3)^100 = 8^100 

Vì 9^100 > 8^100 

Vậy 3^200 > 2^300 

\(3^{20}=\left(3^2\right)^{10}=9^{10}\)

\(2^{30}=\left(2^3\right)^{10}=8^{10}\)

Ta có\(9>8\Rightarrow9^{10}>8^{10}\Rightarrow3^{20}>2^{30}\)

Vậy\(3^{20}>2^{30}\)

Chắc chắn là 3 lũy thừa  30 rồi

nhầm \(3>2\Rightarrow3^{30}>2^{30}\)

Ta co : \(3^{500}\&7^{300}\)

\(\Rightarrow3^{500}=\left(3^5\right)^{100}=243^{100}\)

\(\Rightarrow7^{300}=\left(7^3\right)^{100}=343^{100}\)

Ta thay \(243^{100}

3^500=(3^5)^100=243^100

7^300=(7^3)^100=343^100

3⁵⁰⁰ = 3^5.100 = ( 3⁵)¹⁰⁰ 

7³⁰⁰ = 7^3.100 = ( 7³)¹⁰⁰ 

Vì 3⁵ < 7³ nên => 3⁵⁰⁰ < 7³⁰⁰ 

\(201^{60}=\left(201^4\right)^{15}=1944810000^{15}\)

\(398^{45}=\left(398^3\right)^{15}=63044792^{15}\)

Do \(1944810000>63044792\)

\(\Rightarrow1944810000^{15}>63044792^{15}\)

\(\Rightarrow201^{60}>398^{45}\)

Ta có:

\(201^{60}>200^{60};398^{45}< 400^{45}\)

\(200^{60}=\left(2.100\right)^{60}=2^{60}.100^{60}=2^{60}.\left(10^2\right)^{60}\)

\(=2^{60}.10^{120}=2^{60}.10^{30}.10^{90}\)

\(400^{45}=\left(2.100\right)^{45}=2^{45}.100^{45}=2^{45}.\left(10^2\right)^{45}\)

\(=2^{45}.10^{90}\)

Mà \(2^{60}.10^{30}.10^{90}>2^{45}.10^{90}\)

\(\Rightarrow200^{60}>400^{45}\)

\(\Rightarrow201^{60}>200^{60}>400^{45}>398^{45}\)

\(\Rightarrow201^{60}>398^{45}\)

`#3107`

\(201^{60}\text{ và }398^{45}\)

Ta có:

\(201^{60}=\left(201\right)^{15\cdot4}=\left(201^4\right)^{15}=1632240801^{15}\)

\(398^{45}=\left(398\right)^{15\cdot3}=\left(398^3\right)^{15}=63044792^{15}\)

Vì `63044792 < 1632240801 \Rightarrow`\(1632240801^{15}< 63044792^{15}\)

\(\Rightarrow201^{60}>398^{45}\)

Vậy, \(201^{60}>398^{45}.\)