1 + 2 + 3 + ... + x = 465  (x > 0)

x.(x + 1) : 2 = 465

x.(x + 1) = 465 . 2

x² + x = 930

x² + x - 930 = 0

x² + 31x - 30x - 930 = 0

(x² + 31x) - (30x + 930) = 0

x(x + 31) - 30(x + 31) = 0

(x + 31)(x - 30) = 0

x + 31 = 0 hoặc x - 30 = 0

*) x + 31 = 0

x = -31 (loại)

*) x - 30 = 0

x = 30 (nhận)

Vậy x = 30

\(1+2+3+...+x=465\) (ĐK: \(x\in N\))

\(\Rightarrow\left(x+1\right)\left[\left(x-1\right):1+1\right]:2=465\)

\(\Rightarrow\left(x+1\right)\left(x-1+1\right):2=465\)

\(\Rightarrow\left(x+1\right)\cdot x:2=465\)

\(\Rightarrow x\left(x+1\right)=465\cdot2\)

\(\Rightarrow x\cdot\left(x+1\right)=930\)

Mà: \(30\cdot31=930\)

Nên: \(x=30\) 

Vậy: x=30

\(1\le3^{n+2}\le729\)

\(\Rightarrow3^0\le3^{n+2}\le3^6\)

\(\Rightarrow0\le n+2\le6\)

\(\Rightarrow0-2\le n\le6-2\)

\(\Rightarrow-2\le n\le4\)

Mà: \(n\in N^+\)

\(\Rightarrow0\le n\le4\)

\(\Rightarrow n\in\left\{0;1;2;3;4\right\}\)

Sửa đề:

1 ≤ 3ⁿ⁺² ≤ 729

3⁰ ≤ 3ⁿ⁺² ≤ 3⁶

0 ≤ n + 2 ≤ 6

-2 ≤ n ≤ 4

Do n ∈ ℕ

⇒ n ∈ {0; 1; 2; 3; 4}

A = 1 - 2 + 2² - 2³ + ... + 2²⁰⁰⁶ - 2²⁰⁰⁷

2A = 2 - 2² + 2³ - 2⁴ + ... + 2²⁰⁰⁷ - 2²⁰⁰⁸

3A = A + 2A

= (1 - 2 + 2² - 2³ + ... + 2²⁰⁰⁶ - 2²⁰⁰⁷) + (2 - 2² + 2³ - 2⁴ + ... + 2²⁰⁰⁷ - 2²⁰⁰⁸)

= 1 - 2²⁰⁰⁸

A = (1 - 2²⁰⁰⁸)/3

A = 1 - 2 + 2² - 2³ + ... + 2²⁰⁰⁶ - 2²⁰⁰⁷

2A = 2 - 2² + 2³ - 2⁴ + ... + 2²⁰⁰⁷ - 2²⁰⁰⁸

3A = A + 2A

= (1 - 2 + 2² - 2³ + ... + 2²⁰⁰⁶ - 2²⁰⁰⁷) + (2 - 2² + 2³ - 2⁴ + ... + 2²⁰⁰⁷ - 2²⁰⁰⁸)

= 1 - 2²⁰⁰⁸

A = (1 - 2²⁰⁰⁸)/3

Vậy A=(1 - 2²⁰⁰⁸)/3

\(5\le5^n\le625\) 

\(\Rightarrow5^1\le5^n\le5^4\)

\(\Rightarrow1\le n\le4\)

Mà: \(n\in N^+\)

\(\Rightarrow n=\left\{1;2;3;4\right\}\)

Sửa đề:

5 ≤ 5ⁿ ≤ 625

5 ≤ 5ⁿ ≤ 5⁴

1 ≤ n ≤ 4

n ∈ {1; 2; 3; 4}

\(\dfrac{7^x}{7^2}=343\)

\(\Rightarrow7^{x-2}=343\)

\(\Rightarrow7^{x-2}=7^3\)

\(\Rightarrow x-2=3\)

\(\Rightarrow x=3+2\)

\(\Rightarrow x=5\)

7ˣ/7² = 343

7ˣ = 343 . 7²

7ˣ = 7³.7²

7ˣ = 7⁵

x = 5

\(3^5\cdot3^x=3^7\)

\(\Rightarrow3^{x+5}=3^7\)

\(\Rightarrow x+5=7\)

\(\Rightarrow x=7-5\)

\(\Rightarrow x=2\)

\(3^5.3^x=3^7\)

\(\Rightarrow3^x=3^7:3^5\)

\(\Rightarrow3^x=3^{7-5}\)

\(\Rightarrow3^x=3^2\)

\(\Rightarrow3^x=9\)

\(\Rightarrow3^3=9\)

Vậy \(x=3\)

\(\left(3x-1\right)^5-2=30\)

\(\Rightarrow\left(3x-1\right)^5=32\)

\(\Rightarrow\left(3x-1\right)^5=2^5\)

\(\Rightarrow3x-1=2\)

\(\Rightarrow3x=1+2\)

\(\Rightarrow3x=3\)

\(\Rightarrow x=\dfrac{3}{3}\)

\(\Rightarrow x=1\)

\(2^3\cdot3^5ab^3\cdot\left(2\cdot3^2a^3b^2\right)\)

\(=8\cdot243\cdot ab^3\cdot2\cdot9\cdot a^3b^2\)

\(=1944\cdot18\cdot\left(a\cdot a^3\right)\cdot\left(b^3\cdot b^2\right)\)

\(=34992a^4b^5\)

2³.3⁵ab³.(2.3²a³b²)

= (2³.2).(3⁵.3²).(a.a³).(b³.b²)

= 2⁴.3⁷a⁴b⁵

= 34992a⁴b⁵

a) \(...\Rightarrow\left[\left(x-2\right):2+1\right]\left(x+2\right):2=60\)

\(\Rightarrow\left[\dfrac{x-2}{2}+1\right]\left(x+2\right)=120\)

\(\Rightarrow\dfrac{x}{2}\left(x+2\right)=120\)

\(\Rightarrow x\left(x+2\right)=240\)

\(\Rightarrow x\in\varnothing\left(x\inℕ\right)\)

2+4+6+...+ x=60
(2+ X)x  x/2 =120
(2+X)x X = 120 x 2
2x X+X x X= 120
Vì x là số chẵn nên ta thử như sau
X=10   suy ra X x X+2 x X =100 +20=120  (đúng)
X=8      suy ra X x X+2 x X=64+16=80  (loại)
Suy ra X=10

(nếu X càng nhỏ hơn 8 thì đáp số không thể bằng 120)

a) \(\text{13 x 58 x 4 + 32 x 26 x 2 + 52 x 10 }\)

\(\text{=(13 x 4 ) x 58 + 32 x 13 x 2 x 2 + 13 x 4 x 10}\)

\(\text{= ( 13 x 4 ) x 58 + 32 x ( 13 x 2 x 2 ) + ( 13 x 4 ) x 10 }\)

\(\text{= 52 x 58 + 32 x 52 + 52 x 10}\)

\(\text{=52 x ( 58 + 32 + 10 )}\)

\(\text{=52 x 100 = 5200}\)

b) \(\text{14 x 35 x 5 + 10 x 25 x 7 + 20 x 70 x 2}\)

\(\text{= ( 14 x 5 ) x 35 + (10 x 7 ) x 25 + 20 x 70 x 2 }\)

\(\text{= 70 x 35 + 70 x 25 + 20 x 2 x 70 }\)

\(\text{= 70 x ( 35 + 25 + 40 )}\)

\(\text{= 70 x 100 = 7000 }\)

e) \(\text{5.3.37.4+5.24.21+21.5.12}\)

\(\text{= 5.(3.37.4+24.21+21.12)}\)

\(\text{= 5.1200}\)

\(=6000\)