a) Ta có 31<2⁵=> 31¹¹<(2⁵)¹¹= 2⁵⁵

Lại có 17>2⁴=>17¹⁴>(2⁴)¹⁴= 2⁵⁶ hay 2⁵⁶<17¹⁴

Mà 2⁵⁵<2⁵⁶=>31¹¹<2⁵⁶

Mà 2⁵⁶<17¹⁴ nên 31¹¹<17¹⁴

b)Ta có 2¹⁰⁰=(2¹⁰)¹⁰=1024¹⁰>1000¹⁰=10³⁰

=>2¹⁰⁰>10³⁰

c)Ta có 5³⁶= (5³)¹²=125¹²

           11²⁴= (11²)¹² = 121¹²

Vì 125¹²>121¹²nên 5³⁶>11²⁴

a ) 

 3⁴⁰⁰⁰ và 9²⁰⁰⁰

ta có: 3⁴⁰⁰⁰ = (3⁴)¹⁰⁰⁰ = 81¹⁰⁰⁰

            9²⁰⁰⁰ = (9²)¹⁰⁰⁰ = 81¹⁰⁰⁰

vậy 3⁴⁰⁰⁰ = 9²⁰⁰⁰

b ) (222³)¹¹¹ và  (333²)¹¹¹

(2 x 111)³ và  (3 x 111)²

8 x 111³    và  9 x 111²

888 x 111² và 9 x 111².

Kết luận : 222^333 > 333^222.

c ) 5³⁶ = (5³)¹² = 125¹²

11²⁴ = (11²)¹² = 121¹²

125¹² > 121¹²  nên 5³⁶ > 11²⁴

a. Ta có: \(5^{36}=\left(5^3\right)^{12}=125^{12}\)

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

\(Do\)\(125^{12}>121^{12}\Rightarrow5^{36}>11^{24}\)

b) Ta có; \(6.5^{22}>5.5^{22}=5^{23}\)

d) Ta có: \(72^{45}-72^{44}=72^{44}\left(72-1\right)=72^{44}.71\)

\(72^{44}-72^{43}=72^{43}\left(72-1\right)=72^{43}.71\)

Do \(72^{44}.71>72^{43}.71\Rightarrow72^{45}-72^{44}>72^{44}-72^{43}\)

\(5^{30}=\left(5^3\right)^{10}=125^{10}\)

Vì 125¹⁰>124¹⁰=>5³⁰>124¹⁰

a,

\(5^{30}\)và  \(124^{10}\)

\(5^{30}=\left(5^3\right)^{10}=125^{10}\)

Vì \(125^{10}>124^{10}\)nên \(5^{30}>124^{10}\)

b, \(31^{11}\)và  \(17^{14}\)

\(31^{11}< 32^{11}=\left(2^5\right)^{11}=2^{55}\)

\(17^{14}< 16^{14}=\left(2^4\right)^{14}=2 ^{56}\)

Vì \(2^{55}< 2^{56}\)nên \(31^{11}< 17^{14}\)

a/

2020.2021=(2019+1)(2022-1)=

=2019.2022-2019+2022-1=2019.2022+2>2019.2022

b/

\(4^7=\left(2^2\right)^7=2^{14}< 2^{15}\)

c/

\(199^{20}< 200^{20}=\left(8.25\right)^{20}=\left(2^3.5^2\right)^{20}=2^{60}.5^{40}\)

\(2000^{15}=\left(16.125\right)^{15}=\left(2^4.5^3\right)^{15}=2^{60}.5^{45}\)

\(\Rightarrow2000^{15}=2^{60}.5^{45}>2^{60}.5^{40}>199^{20}\)

d/

\(31^{31}< 32^{31}=\left(2^5\right)^{31}=2^{155}\)

\(17^{39}>16^{39}=\left(2^4\right)^{39}=2^{156}\)

\(\Rightarrow17^{39}=2^{156}>2^{155}>31^{31}\)

31^11 và 17^14

31^11 < 32^11 = ( 2⁵ )¹¹= 2^5.11=2⁵⁵

17^14 > 16^ 14 = ( 2⁴)¹⁴=2^4.14=2⁵⁶

Mà 2⁵⁵<2⁵⁶ Nên 31^11<17^14

31^11 và 17^14

31^11 < 32^11 = ( 2⁵ )¹¹= 2^5.11=2⁵⁵

17^14 > 16^ 14 = ( 2⁴)¹⁴=2^4.14=2⁵⁶

Mà 2⁵⁵<2⁵⁶ Nên 31^11<17^14

\(31^{11}< 32^{11}=\left(2^5\right)^{11}=2^{55}\)

\(17^{14}>16^{14}=\left(2^4\right)^{14}=2^{56}\)

\(\Rightarrow17^{14}>2^{56}>2^{55}>31^{11}\)