a,

Ta có:

16¹⁹ = (2⁴)¹⁹ = 2⁷⁶

8²⁵ = (2³)²⁵ = 2⁷⁵

Vì 2⁷⁶ > 2⁷⁵

Nên 16¹⁹ > 8²⁵.

b,

Ta có :

3^{10 =} 3^{2 .5} = (3²)⁵ = 9⁵

2¹⁵ - 2^{3 .5} = (2³)⁵ = 85
Vì 9^{5 >} 8⁵
Nên 3²⁰ > 2¹⁵

Câu a:

5^3 và 3^5

5^3 = 125

3^5 = 243 > 125

Vậy 5^3 < 3^5

Câu b:

16^19 và 8^25

16^19 = (2^4)^19 = 2^76

8^25 = (2^3)25 = 2^75 < 2^76

Vậy 16^19 > 8^25

Câu c:

27^11 và 81^8

27^11 = (3^3)^11 = 3^33

81^8 = (3^4)^8 = 3^32 < 3^33

Vậy 27^11 > 81^8

Câu 6c:

13^40 và 2^161

2^161 = (2^4)^40.2 = 2.(16^40) > 13^40

Vậy 13^40 < 2^161

Câu 6d:

5^300 và 3^453

5^300 = (5^2)^150 = 25^150

3^453 > 3^450 = (3^3)^150 = 27^150

25^150 < 27^150

Vậy 5^300 < 3^453

a/ \(2A=2+2^2+2^3+2^4+...+2^{2011}\)

\(A=2A-A=2^{2011}-2^0=2^{2011}-1=B\)

b/ \(A=2009.2011=\left(2010-1\right)\left(2010+1\right)=2010^2-1< B=2010^2\)

c/ 

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

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

\(\Rightarrow11^{24}=121^{12}< 125^{12}=5^{36}\)

d/ 

\(625^5=\left(5^4\right)^5=5^{20}\)

\(125^7=\left(5^3\right)^7=5^{21}>5^{20}=625^5\)

e/

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

\(2^{3n}=\left(2^3\right)^n=8^n< 9^n=3^{2n}\)

f/

\(6.5^{22}>5.5^{22}=5^{23}\)

g/

\(333^{444}=\left(3.111\right)^{444}=3^{444}.111^{444}=\left(3^4\right)^{111}.111^{444}=81^{111}.111^{444}\)

\(444^{333}=\left(4.111\right)^{333}=4^{333}.111^{333}=\left(4^3\right)^{111}.111^{333}=64^{111}.111^{333}\)

\(\Rightarrow333^{444}>444^{333}\)

c) 5³⁶  và    11²⁴

5³⁶ = (5³)¹² = 125¹²

11²⁴ = (11²)¹² = 121¹²

Vì 125 > 121 nên 5³⁶ > 11²⁴

d)3²⁰⁰   và   2³⁰⁰

3²⁰⁰ = (3²)¹⁰⁰ = 9¹⁰⁰

2³⁰⁰= (2³)¹⁰⁰ = 8¹⁰⁰

Vì 9 > 8 nên 3²⁰⁰ > 2³⁰⁰

e) 3²ⁿ và 2³ⁿ

3²ⁿ = (3²)ⁿ = 9ⁿ

2³ⁿ = (2³)ⁿ = 8ⁿ

Vì 9 > 8 nên 3²ⁿ > 2³ⁿ

b)  625²⁵   và 125³⁰

625²⁵ = ( 5⁴)²⁵ = 5¹⁰⁰

125³⁰ = ( 5³ )³⁰ = 5⁹⁰

Vì 100 > 90 nên 625²⁵ > 125³⁰

\(a;5^{23}=5\cdot5^{22}< 6\cdot5^{22}\Rightarrow5^{23}< 6\cdot5^{22}\)

\(b;7\cdot2^{13}< 8\cdot2^{13}=2^3\cdot2^{13}=2^{15}\)

\(c;21^{15}=3^{15}\cdot7^{15}>3^{15}\cdot7^{14}=27^5\cdot49^8\)

\(d;199^{20}< 200^{20}=10^{40}\cdot2^{20}< 10^{45}\cdot2^{15}=2000^{15}< 2001^{15}\)

\(e;3^{39}=9^{13}< 11^{13}< 11^{21}\)

a) 27¹¹ = (3³)¹¹ = 3³³; 81⁸ = (3⁴)⁸ = 3³²

Do 33 > 32 => 27¹¹ > 81⁸

b) 16¹⁹ = (2⁴)¹⁹ = 2⁷⁶; 8²⁵ = (2³)²⁵ = 2⁷⁵

Do 76 > 75 => 16¹⁹ > 8²⁵

c) 625⁵ = (5⁴)⁵ = 5²⁰; 125⁷ = (5³)⁷ = 5²¹

Do  20 < 21 => 625⁵ < 125⁷

d) 5³⁶ = (5³)¹² = 125¹²; 11²⁴ = (11²)¹² = 121¹²

Do 125 > 121 => 5³⁶ > 11²⁴

e) 3²ⁿ = 9ⁿ; 2³ⁿ = 8ⁿ; Do 9 > 8 => 3²ⁿ > 2³ⁿ

f) 3⁵⁴ = (3²)²⁷ = 9²⁷; 2⁸¹ = (2³)²⁷ = 8²⁷

Do 9 > 8 => 3⁵⁴ > 2⁸¹

b) \(625^5=\left(5^4\right)^5=5^{20}< 125^7=\left(5^3\right)^7=5^{21}\)

e) \(5^{23}=5.5^{22}< 6.5^{22}\)