a/ \(2^{225}=\left(2^3\right)^{75}=8^{75}\)

\(3^{151}>3^{150}=\left(3^2\right)^{75}=9^{75}\)

Mà \(8^{75}< 9^{75}\)

=> \(2^{225}< 3^{150}< 3^{151}\)

b/ Xét n là số lẻ

=> n + 1 chẵn

=> n + 1 ⋮ 2

=> (n+1)(3n+2) ⋮2

Xét n là số chẵn

=> 3n chẵn

=> 3n+2 chẵn

=> (n+1)(3n+2) ⋮2

Do đó A = (n+1)(3n+2) chia hết cho 2 với mọi số tự nhiên n 

\(1,\\ a,2^x=16=2^4\Rightarrow x=4\\ b,3^{x+1}=9^x=3^{2x}\\ \Rightarrow x+1=2x\Rightarrow x=1\\ c,2^{3x+2}=4^{x+5}=2^{2\left(x+5\right)}\\ \Rightarrow3x+2=2x+10\Rightarrow x=8\\ d,3^{2x-1}=243=3^5\\ \Rightarrow2x-1=5\Rightarrow x=3\\ 2,\\ a,2^{225}=8^{75}< 9^{75}=3^{150}\\ b,2^{91}=\left(2^{13}\right)^7=8192^7>3125^7=\left(5^5\right)^7=5^{35}\\ c,99^{20}=\left(99^2\right)^{10}< \left(99\cdot101\right)^{10}=9999^{10}\\ 3,\\ a,12^8\cdot9^{12}=2^{16}\cdot3^8\cdot3^{24}=2^{16}\cdot3^{32}=\left(2\cdot3^2\right)^{16}=18^{16}\\ b,75^{20}=\left(3\cdot5^2\right)^{20}=3^{20}\cdot5^{40}=\left(3^{20}\cdot5^{10}\right)\cdot5^{30}=\left(3^2\cdot5\right)^{10}\cdot5^{30}=45^{10}\cdot5^{30}\)

Bài 1:

a) \(\Rightarrow2^x=2^4\Rightarrow x=4\)

b) \(\Rightarrow3^{x+1}=3^{2x}\Rightarrow x+1=2x\Rightarrow x=1\)

c) \(\Rightarrow2^{3x+2}=2^{2x+10}\Rightarrow3x+2=2x+10\Rightarrow x=8\)

d) \(\Rightarrow3^{2x-1}=3^5\Rightarrow2x-1=5\Rightarrow x=3\)

Bài 2:

a) \(2^{225}=\left(2^3\right)^{75}=8^{75}< 9^{75}=\left(3^2\right)^{75}=3^{150}\)

b) \(2^{91}=\left(2^{13}\right)^7=8192^7>3125^7=\left(5^5\right)^7=5^{35}\)

c) \(99^{20}=\left(99^2\right)^{10}=9801^{10}< 9999^{10}\)

Bài 3:

a) \(12^8.9^{12}=\left(4.3\right)^8.9^{12}=4^8.3^8.9^{12}=2^{16}.9^4.9^{12}=2^{16}.9^{16}=\left(2.9\right)^{16}=18^{16}\)

b) \(75^{20}=\left(75^2\right)^{10}=5625^{10}=\left(45.125\right)^{10}=45^{10}.125^{10}=45^{10}.5^{30}\)

2225 = 23.75 = (23)75 = 875

3150 = 32.75 = (32)75=975

8 < 9 ⇒ 875 < 975

Vậy : 2225 < 3150

1 /

A = B

2 / 

A = 2^300 = (2^3)^100 = 8^100

B = 3^200 = ( 3^2)^100 = 9^100

Vì 8^100 < 9^100 nên A < B

27⁵ và 243³

ta có: 

27⁵ = (3³)⁵ = 3^3.5 =3¹⁵

243³= (3⁵)³ = 3¹⁵

Vì 3¹⁵ = 3¹⁵ => A=B

2³⁰⁰ và 3²⁰⁰

ta có: 2³⁰⁰ = (2³)¹⁰⁰ = 9¹⁰⁰

3²⁰⁰  = (3²)¹⁰⁰ = 9¹⁰⁰

Vì 9¹⁰⁰ = 9¹⁰⁰ => A=B

SỬA LẠI 2³⁰⁰  và 3²⁰⁰

2³⁰⁰ = (2³)¹⁰⁰ = 8¹⁰⁰

3²⁰⁰ = 9¹⁰⁰ (cách biến đổi như ở cmt dưới)

=> A<B

a) – 4 < |– 4|;                  b ) 15 = |–15|;                c) |– 31|  >  |– 16|; d) |– 5| > 0;           e)  |–2| > 0

a) \(6=\sqrt[3]{6^3}=\sqrt{216}>\sqrt[3]{208}=2\sqrt[3]{26}\)

b) \(2\sqrt[3]{6}=\sqrt[3]{2^3.6}=\sqrt[3]{48}>\sqrt[3]{47}\)