Ta có:

\(A=1+2+2^2+2^3+...+2^{2021}+2^{2022}\)

\(\Rightarrow2A=2\left(1+2+2^2+...+2^{2022}\right)\)

\(\Rightarrow2A=2+2^3+2^4+...+2^{2023}\)

\(\Rightarrow2A-A=\left(2+2^2+2^3+...+2^{2023}\right)-\left(1+2+2^2+...+2^{2022}\right)\)

\(\Rightarrow A=2^{2023}-1\)

Ta thấy: \(2^{2023}-1=2^{2023}-1\)

Vậy: \(A=B\)

Bài 1: 

a) 0²⁰⁰² < 0²⁰²³

b) 2022⁰ = 2023⁰

c) 54⁹ < 55¹⁰

d) ( 4 + 5 )³ > 4² + 5²

đ) 9² - 3² > ( 9 - 3 )²

Bài 2:

a) 3² x 4³ - 3² + 333

= 9 x 64 - 9 + 333

= 576 - 9 + 333

= 567 + 333

= 900

b) 5 x 4³ + 24 x 5 + 41⁰

= 5 x 64 + 24 x 5 + 1

= 5 x ( 64 + 24 ) + 1

= 5 x 88 + 1

= 440 + 1

= 441

c) 2³ x 4² + 3² x 5 - 40 x 1²⁰²³

= 8 x 16 + 9 x 5 - 40 x 1

= 128 + 45 - 40

= 133

Bài 1 :

a) \(0^{2002}=0;0^{2023}=0\Rightarrow0^{2002}=0^{2023}\)

b) \(2022^0=1;2023^0=1\Rightarrow2022^0=2023^0\)

c) \(54^9< 55^9;55^9< 55^{10}\Rightarrow54^9< 55^{10}\)

d) \(\left(4+5\right)^3>\left(4+5\right)^2;\left(4+5\right)^2>4^2+5^2\Rightarrow\left(4+5\right)^3>4^2+5^2\)

đ) \(9^2-3^2=81-9=82;\left(9-3\right)^2=6^2=36\Rightarrow9^2-3^2>\left(9-3\right)^2\)

Bài 2 :

a) \(3^2.4^3-3^2+333=3^2\left(4^3-1\right)+9.37=9.63+9.37=9\left(63+37\right)=9.100=900\)

b) \(5.4^3+24.5+41^0=20.4^2+20.6+1=20\left(16+6\right)+1=20.22+1=441\)

c) \(2^3.4^2+3^2.5-40.1^{2023}=8.16+9.5-40.1=128+45-40=128+5=133\)

Áp dụng tính chất : Nếu \(\dfrac{a}{b}< 1\) thì \(\dfrac{a}{b}< \dfrac{a+n}{b+n}\) ( a; b; n ϵ N , b; n ≠ 0 )

Ta có \(\dfrac{2023^{31}+5}{2023^{32}+5}< 1\)

⇒ \(B=\dfrac{2023^{31}+5}{2023^{32}+5}< \dfrac{2023^{31}+5+2018}{2023^{32}+5+2018}=\dfrac{2023^{31}+2023}{2023^{32}+2023}=\dfrac{2023\left(2023^{30}+1\right)}{2023\left(2023^{31}+1\right)}=\dfrac{2023^{30}+1}{2023^{31}+1}=A\)Vậy A > B

Ta có 2023A = \(\dfrac{2023.\left(2023^{30}+5\right)}{2023^{31}+5}=\dfrac{2023^{31}+5.2023}{2023^{31}+5}\)

\(=1+\dfrac{2022.5}{2023^{31}+5}\)

Lại có 2023B = \(\dfrac{2023.\left(2023^{31}+5\right)}{2023^{32}+5}=\dfrac{2023^{32}+2023.5}{2023^{32}+5}\)

\(=1+\dfrac{2022.5}{2023^{32}+5}\)

Dễ thấy 2023³¹ + 5 < 2023³² + 5

\(\Leftrightarrow\dfrac{2022.5}{2023^{31}+5}>\dfrac{2022.5}{2023^{32}+5}\)

\(\Leftrightarrow1+\dfrac{2022.5}{2023^{31}+5}>1+\dfrac{2022.5}{2023^{32}>5}\)

\(\Leftrightarrow2023A>2023B\Leftrightarrow A>B\)

\(3B=1.3^2+2.3^3+3.3^4+...+2022.3^{2023}+2023.3^{2024}\)

\(2B=3B-B=-3-3^2-3^3-...-3^{2023}+2023.3^{2024}\)

\(2B=2023.3^{2024}-\left(3+3^2+3^3+...+3^{2023}\right)\)

Đặt 

\(C=3+3^2+3^3+...+3^{2023}\)

\(3C=3^2+3^3+3^4+...+3^{2024}\)

\(2C=3C-C=3^{2024}-3\Rightarrow C=\dfrac{3^{2024}-3}{2}\)

\(\Rightarrow2B=2023.3^{2024}-\dfrac{3^{2024}-3}{2}=\)

\(=\dfrac{2.2023.3^{2024}-3^{2024}+3}{2}=\dfrac{4045.3^{2024}+3}{2}\)

\(\Rightarrow B=\dfrac{4045.3^{2024}+3}{4}\)

Lời giải:

$A=9+2.3^2+2.3^3+2.3^4+...+2.3^{2023}$

$A-9=2(3^2+3^3+3^4+...+3^{2023})$

$3(A-9)=2(3^3+3^4+3^5+...+3^{2024})$

$\Rightarrow 3(A-9)-(A-9)=2(3^{2024}-3^2)$

$2(A-9)=2.3^{2024}-18$

$\Rightarrow 2A-18=2.3^{2024}-18$

$\Rightarrow A=3^{2024}\vdots 3^{2023}$ (đpcm)

Bạn nên viết lại đề bằng công thức toán (biểu tượng $\sum$ góc trái khung soạn thảo) để mọi người hiểu đề của bạn hơn nhé.

chịu

A = |\(x\) + 5| + 2023

|\(x\) + 5| ≥ 0 ⇒| \(x\) + 5| + 2023 ≥ 2023⇒ A(min) = 2023 xảy ra khi \(x\) = -5

B = (\(x+2\))² - 2023

(\(x\) + 2)2 ≥ 0 ⇒ (\(x\) + 2)2 ≥ - 2023 ⇒ A(min) = -2023  xảy ra khi \(x\) = -2

C = \(x^2\) - 6\(x\) + 20

C = (\(x^2\) - 3\(x\)) - ( 3\(x\) - 9) + 11

C = \(x\)(\(x-3\)) - 3(\(x\) -3) + 11

C = (\(x-3\))(\(x\)-3) + 11

C = (\(x-3\))² + 11

(\(x\) -3)² ≥ 0 ⇒ (\(x\) - 3)² + 11 ≥ 11 vậy C(min) = 11 xảy ra khi \(x=3\)

D = \(x^2\) + 10\(x\) - 25

D = \(x^2\) + 5\(x\) + 5\(x\) + 25 - 55

D = (\(x^2\) + 5\(x\)) + (5\(x\) + 25) - 50

D = \(x\)(\(x\) + 5) + 5(\(x\) + 5)  - 50

D = (\(x\) +5)(\(x\) + 5) - 50

D = ( \(x\) + 5)² - 50

(\(x+5\))² ≥ 0 ⇒ (\(x\) + 5)² - 50 ≥ -50 ⇒ D(min) = -50 xảy ra khi \(x\) = -5