a) 10^30 và 2^100
Ta có: 10^30 = (10^3)^10 = 1000^10
          2^100 = (2^10)^10 = 1024^10
Do 1024^10 > 1000^10 => 2^100 > 10^30

b) 333^444 và 444^333
Ta có: 333^444 = 111^444 x 3^444 
          444^333 = 111^333 x 4^333 
Tách: 3^444 = (3^4)^111 =81^111 <=>4^333 = (4^3)^111 = 64^111 
Mà: {111^444 > 111^333 (1) 
       {81^111 > 64^111 hay: (3^4)^111 > (4^3)^111 (2) 
Từ (1) và (2) ta có:333^444 > 444^333

c) 3^450 =(3^3)^150 =27^150 
5^300=(5^2)^150=25^150 
vì 27^150 >25^150 =>3^450 > 5^300 
vậy 3^450 > 5^300

a) \(10^{30}=\left(10^3\right)^{10}=1000^{10}\)

\(2^{100}=\left(2^{10}\right)^{10}=1024^{10}\)

Mà \(1000^{10}< 1024^{10}\Rightarrow10^{30}< 2^{100}\)

b) \(3^{400}=\left(3^4\right)^{100}=81^{100}\)

\(5^{300}=\left(5^3\right)^{100}=125^{100}\)

Mà \(81^{100}< 125^{100}\Rightarrow3^{400}< 5^{300}\)

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

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

Mà \(81^{111}.111^{444}>64^{111}.111^{333}\Rightarrow333^{444}>444^{333}\)

444^333

a) Do 300 < 450

⇒ 3³⁰⁰ < 3⁴⁵⁰

b) 333⁴⁴⁴ = (333⁴)¹¹¹ = (111⁴.3⁴)¹¹¹

444³³³ = (444³)¹¹¹ = (111³.4³)¹¹¹

Do 4 > 3 nên 111⁴ > 111³ (1)

Lại có:

3⁴ = 81

4³ = 64

Do 81 > 64 nên 3⁴ > 4³ (2)

Từ (1) và (2) ⇒ 111⁴.3⁴ > 111³.4³

⇒ (111⁴.3⁴)¹¹¹ > (111³.4³)¹¹¹

Vậy 333⁴⁴⁴ > 444³³³

                                                        Bài giải

a, \(3^{450}=\left(3^3\right)^{150}=9^{150}\)

\(5^{300}=\left(5^2\right)^{150}=25^{150}\)

\(\text{Vì }9^{150}< 25^{150}\) \(\Rightarrow\text{ }3^{450}< 5^{300}\)

b, \(333^{444}=\left(333^4\right)^{111}=12296370321^{111}\)

\(444^{333}=\left(444^3\right)^{111}=87528384^{111}\)

Vì  \(12296370321^{111}>87528384^{111}\) \(\Rightarrow\text{ }333^{444}>444^{333}\)

                                                        Bài giải

a, \(3^{450}=\left(3^3\right)^{150}=9^{150}\)

\(5^{300}=\left(5^2\right)^{150}=25^{150}\)

\(\text{Vì }9^{150}< 25^{150}\) \(\Rightarrow\text{ }3^{450}< 5^{300}\)

b, \(333^{444}=\left(333^4\right)^{111}=12296370321^{111}\)

\(444^{333}=\left(444^3\right)^{111}=87528384^{111}\)

Vì  \(12296370321^{111}>87528384^{111}\) \(\Rightarrow\text{ }333^{444}>444^{333}\)

C=3⁴⁵⁰ và D=5³⁰⁰

C=3⁴⁵⁰=(3³)¹⁵⁰=27¹⁵⁰

D=5³⁰⁰=(5²)¹⁵⁰=25¹⁵⁰

Vì C=27¹⁵⁰>D=25¹⁵⁰

Nên:C=3⁴⁵⁰>D=5³⁰⁰

E=333⁴⁴⁴ và F=444³³³

E=333⁴⁴⁴ = (3.111)^4.111 = (81.111⁴)¹¹¹

F=444³³³ = (4.111)^3.111 = (64.111³)¹¹¹

Vì E=(81.111⁴)¹¹¹ > F(64.111³)¹¹¹ nên E=333⁴⁴⁴ > F=444³³³

* 3⁴⁵⁰ và 5³⁰⁰           

ta có: 3⁴⁵⁰=3^3.150=(3³)¹⁵⁰=27¹⁵⁰

5³⁰⁰=5^2.150=(5²)¹⁵⁰=25¹⁵⁰

vì 27>25 nên 27¹⁵⁰>25¹⁵⁰

hay 3⁴⁵⁰>5³⁰⁰

*333⁴⁴⁴ và 444³³³

ta có: 333⁴⁴⁴=(3.111)^4.111=(3⁴.111⁴)¹¹¹

                   =(81.111⁴)¹¹¹

444³³³=(4.111)^3.111=(4³.111³)¹¹¹

          =(64.111³)¹¹¹

vì 81>64 ; 111⁴>111³ nên (81.111⁴)¹¹¹>(64.111³)¹¹¹

hay 333⁴⁴⁴>444³³³

a. 3⁴⁵⁰ = (3³)¹⁵⁰ = 27¹⁵⁰;

5³⁰⁰ = (5²)¹⁵⁰ = 25¹⁵⁰

Vì 27¹⁵⁰ > 25¹⁵⁰

=> 3⁴⁵⁰ > 5³⁰⁰.

b. 333⁴⁴⁴ = (3.111)⁴⁴⁴ = 3⁴⁴⁴.111⁴⁴⁴ =(3⁴)¹¹¹.111⁴⁴⁴=81¹¹¹.111⁴⁴⁴

444³³³=(4.111)³³³=4³³³.111³³³=(4³)¹¹¹.111³³³=64¹¹¹.111³³³

Vì 81¹¹¹ > 64¹¹¹ và 111⁴⁴⁴ > 111³³³

=> 81¹¹¹.111⁴⁴⁴ > 64¹¹¹.111³³³

=> 333⁴⁴⁴ > 444³³³.

c. 2014.2016

= 2014.(2015+1)

= 2014.2015+2014 (1)

2015²

=2015.2015

=2015.(2014+1)

=2015.2014+2015 (2)

Từ (1) và (2) => 2014.2016 < 2015².

b) 333\(^{444}\)và 444\(^{333}\)

Ta có :333\(^{444}\)(3.111)\(^{4.111}\)=(3\(^4\).111\(^4\))\(^{111}\)=(81.111\(^4\)).111

444\(^{333}\)(4.111)\(^{3.111}\)=4\(^3\).111\(^2\))\(^{111}\)=(64.111\(^3\))\(^{111}\) 

vì 81>64 ; 111\(^4\)>111\(^3\) nêb (81.111\(^4\))\(^{111}\)>(64.113\(^3\))\(^{111}\)

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