\(333^{444}=\left(333\times4\right)^{111}=1332^{111}\)

\(444^{333}=\left(444\times3\right)^{111}=1332^{111}\)

\(1332^{111}=1332^{111}\Rightarrow333^{444}=444^{333}\)

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

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

\(\Rightarrow333^4=111^4.3^4=111^3.111.3^4\)

     \(444^3=111^3.4^3\)

\(\Rightarrow111.3^4=111.81>4^3=64\)

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

Ta có: 333⁴⁴⁴=333^111.4=333⁴mũ 111=12296370321¹¹¹

              444³³³=444^111.3=444³mũ 111=87528384¹¹¹

Mà: 12296370321>87528384 và 111=111.

=>333⁴⁴⁴>444³³³.

Tk phát nhé

333^444 = (333^4)^111

444^333 = (444^3)^111

333^4 = 3^4.111^4 = 81.111^4

444^3 = 4^3.111^3 = 64.111^3 < 81.111^4

444^3< 333^4

(333^4)^111 > (444^3)^111

333^444 > 444^333

Câu 2:

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

Câu 3:

333^444 = (333^4)^111

444^333 = (444^3)^111

333^4 = 3^4.111^4 = 81.111^4

444^3 = 4^3.111^3 = 64.111^3 < 81.111^4

444^3< 333^4

(333^4)^111 > (444^3)^111

333^444 > 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 

So sánh A và B ta thấy:

 A<B

 Đ/s:..............

**** nha

Câu d:

333^444 = (333^4)^111

444^333 = (444^3)^111

333^4 = 3^4.111^4 = 81.111^4

444^3 = 4^3.111^3 = 64.111^3 < 81.111^4

444^3< 333^4

(333^4)^111 > (444^3)^111

333^444 > 444^333

Câu e:

13^40 và 12^161

12^161 > 12^160 = (12^4)^40 = 20736^40 > 13^40

Vậy 13^40 < 12^161

C = 2³³³ = (2³)¹¹¹ = 8¹¹¹

D = 3²²² = (3²)¹¹¹ = 9¹¹¹

Vì  8¹¹¹ < 9¹¹¹

Nên : C < D 

1) A=333⁴⁴⁴ và B=444³³³

Ta có 333⁴⁴⁴=(3.111)^4.111=3^4.111.111^4.111=(3⁴.111⁴)¹¹¹=(81.111⁴)¹¹¹

444³³³=(4.111)^3.111=4^3.111.111^3.111=(4³.111³)¹¹¹=(64.111³)¹¹¹

Vì (81.111⁴)¹¹¹> (64.111³)¹¹¹

Nên 333⁴⁴⁴ > 444³³³

2) C=2³³³ và D=333²²²

Ta có 2³³³=2^3.111=(2³)¹¹¹=8¹¹¹

3²²²=3^2.111=(3²)¹¹¹=9¹¹¹

Vì 8¹¹¹ <9¹¹¹

Nên 2³³³< 3²²²

a)\(9^{12}=\left(3^2\right)^{12}=3^{24}\)

 \(27^7=\left(3^3\right)^7=3^{21}\)

\(\Rightarrow9^{12}>27^7\)

a) bạn Mạnh làm rồi và đúng

b) Ta có : \(333^{444}=\left(333^4\right)^{111}=\left[\left(3.111\right)^4\right]^{111}=\left[\left(3^4.111^4\right)\right]^{111}=\left(84.111^4\right)^{111}\)

                \(444^{333}=\left(444^3\right)^{111}=\left[\left(4.111\right)^3\right]^{111}=\left[\left(4^3.111^3\right)\right]^{111}=\left(64.111^3\right)^{111}\)

Ta thấy (84.111⁴)¹¹¹ > ( 64.111³)¹¹¹ => 333⁴⁴⁴ > 444³³³

Vậy...

c) Vì \(17^{2002}+1>17^{2001}+1\)

\(\Rightarrow\frac{17^{2001}+1}{17^{2002}+1}< \frac{17^{2001}+1}{17^{2001}+1}\)

Câu c:

10^30 và 2^100

10^30 = (10^3)^10 = 1000^10

2^100 = (2^10)^10 = 2024^10 > 1000^10

2^100 > 10^30

333^444 = (333^4)^111

444^333 = (444^3)^111

333^4 = 3^4.111^4 = 81.111^4

444^3 = 4^3.111^3 = 64.111^3 < 81.111^4

444^3< 333^4

(333^4)^111 > (444^3)^111

333^444 > 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 

> 

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