25¹¹ > 125⁷

3⁵⁴ > 2⁸¹

3mu 54  < 2 mu  81

3^54=(3^2)279^27

2^81=(2^3)27=8^27

vì 9>8=> 9^27>9^27

vậy 3^54>2^81

Ta có :

\(3^{54}=\left(3^2\right)^{27}=9^{27}\)

\(2^{81}=\left(2^3\right)^{27}=8^{27}\)

Vì \(9^{27}>8^{27}\)\(\Rightarrow\)\(3^{54}>2^{81}\)

Vậy \(3^{54}>2^{81}\)

_Chúc bạn học tốt_

a,

15^12=(3*5)^12=3^12*5^12

81^3*125^5=(3^4)^3*(5^3)^5=3^12*5^15

Vì 12<15 suy ra 5^12<5^15

Suy ra 3^12*5^12<3^12*5^15

\(a.81^3.125^5=\left(3^4\right)^3.\left(5^3\right)^5=3^{12}.5^{15}=3^{12}.5^{12}.5^3=\left(3.5\right)^{12}.5^3=15^{12}.5^3>15^{12}\)

\(b.4^{20}.81^{12}=\left(2^2\right)^{20}.\left(9^2\right)^{12}=2^{40}.9^{24}=2^{20}.2^{20}.9^{20}.9^4=\left(2.9\right)^{20}.2^{20}.9^4=18^{20}.2^{20}.9^4>18^{20}\)

\(c.73^{75}=\left(73^3\right)^{25}=389017^{25}\)

\(107^{50}=107^{2.50}=\left(107^2\right)^{25}=11449^{25}\)

Vì \(389017^{25}>11449^{25}\Rightarrow73^{75}>107^{50}\)

`#3107.101107`

a)

`64^150` và `4^450`

Ta có:

`64^150 = (4^3)^150 = 4^(3*150) = 4^450`

Vì `450 = 450 => 4^450 = 4^450 => 64^150 = 4^450`

Vậy, `64^150 = 4^450`

b)

`81^64` và `27^100`

Ta có:

`81^64 = (3^4)^64 = 3^(4*64) = 3^256`

`27^100 = (3^3)^100 = 3^(3*100) = 3^300`

Vì `256 < 300 => 3^256 < 3^300 => 81^64 < 27^100`

Vậy, `81^64 < 27^100`

c)

`125^1000` và `25^3000`

Ta có:

`125^1000 = (5^3)^1000 = 5^(3*1000) = 5^3000`

Vì `5 < 25 => 5^3000 < 25^3000 => 125^1000 < 25^3000`

Vậy, `125^1000 < 25^3000`

d)

`4^30` và `3^40`

Ta có:

`4^30 = 4^(3*10) = (4^3)^10 = 64^10`

`3^40 = 3^(4*10) = (3^4)^10 = 81^10`

Vì `64 < 81 => 64^10 < 81^10 => 4^30 < 3^40`

Vậy, `4^30 < 3^40`

m)

`2^5000` và `5^2000`

Ta có:

`2^5000 = 2^(5*1000) = (2^5)^1000 = 32^1000`

`5^2000 = 5^(2*1000) = (5^2)^1000 = 25^1000`

Vì `32 > 25 => 32^1000 > 25^1000 => 2^5000 > 5^2000`

Vậy, `2^5000 > 5^2000`

h)

`6^450` và `3^750`

Ta có:

`6^450 = 6^(150*3) = (6^3)^150 = 216^150`

`3^750 = 3^(150*5) = (3^5)^150 = 243^150`

Vì `216 < 243 => 216^150 < 243^150 => 6^450 < 3^750`

Vậy, `6^450 < 3^750`

0)

`333^444` và `444^333`

Ta có:

`333^444 = 333^(4*111) = (333^4)^111 = (3^4 *111^4)^111 = 81^111 * 111^444`

`444^333 = 444^(3*111) = (444^3)^111 = (4^3 * 111^3)^111 = 64^111 * 111^333`

Vì `81 > 64;` `111^444 > 111^333`

`=> 81^111 * 111^444 > 64^111 * 111^333`

Vậy, `333^444 > 444^333.`

a) Ta có:

\(64^{150}=\left(2^6\right)^{150}=2^{900}\)

\(4^{450}=\left(2^2\right)^{450}=2^{900}\)

Mà: \(2^{900}=2^{900}\Rightarrow64^{150}=4^{450}\)

b) Ta có:

\(81^{64}=\left(3^4\right)^{64}=3^{256}\)

\(27^{100}=\left(3^3\right)^{100}=3^{300}\)

Mà: \(3^{300}>3^{256}\Rightarrow27^{100}>81^{64}\)

c) Ta có: 

\(125^{1000}=\left(5^3\right)^{1000}=5^{3000}\)

Mà: \(25^{3000}>5^{3000}\Rightarrow25^{3000}>125^{1000}\)

d) Ta có:

\(4^{30}=\left(4^3\right)^{10}=64^{10}\)

\(3^{40}=\left(3^4\right)^{10}=81^{10}\)

Mà: \(81^{10}>64^{10}\Rightarrow3^{40}>4^{30}\)

m) Ta có:

\(2^{5000}=\left(2^5\right)^{1000}=32^{1000}\)

\(5^{2000}=\left(5^2\right)^{1000}=25^{1000}\)

Mà: \(25^{1000}< 32^{1000}\Rightarrow2^{5000}>5^{2000}\)

h) Ta có:

\(6^{450}=\left(6^3\right)^{150}=216^{150}\)

\(3^{750}=\left(3^5\right)^{150}=243^{150}\)

Mà: \(243^{150}>216^{150}\Rightarrow3^{750}>6^{450}\)

.... 

a) 4⁴⁵⁰ = (4³)¹⁵⁰ = 64¹⁵⁰

b) 81⁶⁴ = (3⁴)⁶⁴ = 3²⁵⁶

27¹⁰⁰ = (3³)¹⁰⁰ = 3³⁰⁰

Do 256 < 300 nên 3²⁵⁶ < 3³⁰⁰

Vậy 81⁶⁴ < 27¹⁰⁰

c) 125¹⁰⁰⁰ = (5³)¹⁰⁰⁰ = 5³⁰⁰⁰

Do 5 < 25 nên 5³⁰⁰⁰ < 25³⁰⁰⁰

Vậy 125¹⁰⁰⁰ < 25³⁰⁰⁰

d) 4³⁰ = (4³)¹⁰ = 64¹⁰

3⁴⁰ = (3⁴)¹⁰ = 81¹⁰

Do 64 < 81 nên 64¹⁰ < 81¹⁰

Vậy 4³⁰ < 3⁴⁰

m) 2⁵⁰⁰⁰ = (2⁵)¹⁰⁰⁰ = 32¹⁰⁰⁰

5²⁰⁰⁰ = (5²)¹⁰⁰⁰ = 25¹⁰⁰⁰

Do 32 > 25 nên 32¹⁰⁰⁰ > 25¹⁰⁰⁰

Vậy 2⁵⁰⁰⁰ > 5²⁰⁰⁰

h) 6⁴⁵⁰ = (6³)¹⁵⁰ = 216¹⁵⁰

3⁷⁵⁰ = (3⁵)¹⁵⁰ = 243¹⁵⁰

Do 216 < 243 nên 216¹⁵⁰ < 243¹⁵⁰

Vậy 6⁴⁵⁰ < 3⁷⁵⁰

o) 333⁴⁴⁴ = (333⁴)¹¹¹ = [(3.111)⁴]¹¹¹ = (3⁴.111⁴)¹¹¹ = (81.111⁴)¹¹¹

444³³³ = (444³)¹¹¹ = [(4.111)³]¹¹¹

= (4³.111³)¹¹¹ = (64.111³)¹¹¹

Do 81 > 64 ⇒ 81.111⁴ > 64.111⁴ (1)

Do 4 > 3 ⇒ 64.111⁴ > 64.111³ (2)

Từ (1) và (2) ⇒ 81.111⁴ > 64.111³

⇒ (81.111⁴)¹¹¹ > (64.111³)¹¹¹

Vậy 333⁴⁴⁴ > 444³³³

Ta có : \(2^{81}>2^{80}\)

\(=2^{80}\)

\(=2^{20.4}\)

\(=\left(2^4\right)^{20}\)

\(=16^{20}>15^{20}\)

\(\Rightarrow2^{81}>15^{20}\)

Vậy \(2^{81}>15^{20}\)

a) 27¹¹ và 81⁸

\(27^{11}=\left(3^3\right)^{11}=3^{3.11}=3^{33}\)

\(81^8=\left(3^4\right)^8=3^{4.8}=3^{32}\)

Vì 3³³ > 3³² ⇒ 27¹¹ >81⁸

b) 5²³ và 6 . 5²²

\(5^{23}=5^{22}.5\)

Vì 5²² . 5 < 6 . 5²² ⇒ 5²³ < 6 . 5²²

Chứng tỏ (a + 2021) - (a + 222) là bội của 2 a thuộc N

a/ \(27^{11}=\left(3^3\right)^{11}=3^{33}\); \(81^8=\left(3^4\right)^8=3^{32}< 3^{33}\Rightarrow81^8< 27^{11}\)

b/ \(3^{2n}=\left(3^2\right)^n=9^n\); \(2^{3n}=\left(2^3\right)^n=8^n< 9^n\Rightarrow2^{3n}< 3^{2n}\)

a. 27¹¹= (3³)¹¹ = 3³³

    81⁸ = (3⁴)⁸ = 3³²

Suy ra 3³³>3³² hay 27¹¹>81⁸

b. 3²ⁿ = (3²)ⁿ = 9ⁿ

     2³ⁿ = (2³)ⁿ = 8ⁿ

Mà 9>8 suy ra 9ⁿ>8ⁿ hay 3²ⁿ>2³ⁿ

c. 5²³ = 5²² . 5

   (6.5)²² = 6²² . 5²²

Vì 6²²>5 suy ra 5²² . 5<6²² . 5²² hay 5²³<(6.5)²²

d. 72⁴⁵-72⁴⁴ = 72⁴⁴(72-1) = 72⁴⁴ . 71

    72⁴⁴-72⁴³ = 72⁴³(72-1) = 72⁴³ . 71

Vì 72⁴⁴>72⁴³ suy ra 72⁴⁴ . 71>72⁴³ . 71 hay 72⁴⁵-72⁴⁴>72⁴⁴-72⁴³

a​)Ta có: 27¹¹=(3³)¹¹=3³³

              81⁸=(3⁴)⁸​=3³²

​vậy 81⁸<27¹¹

\(10^{30}vs\)\(2^{100}\)

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

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

Vì \(1000^{10}< 1024^{10}=>10^{30}< 2^{100}\)

\(3^{54}vs2^{81}\)

\(3^{54}=\left(3^6\right)^9=729^9\)

\(2^{81}=\left(2^9\right)^9=512^9\)

Vì \(729^9>512^9=>3^{54}>2^{81}\)

\(3^{200}=\left(3^2\right)^{100}=9^{100}\)

\(2^{300}=\left(2^3\right)^{100}=8^{100}\)

Vì \(8^{100}< 9^{100}=>2^{300}< 3^{200}\)