1/2 mũ n = 1/2 mũ 6

Vậy x = 6

\(\left(\dfrac{1}{2}\right)^n=\left(\dfrac{1}{8}\right)^2\)

\(=>\left(\dfrac{1}{2}\right)^n=\left[\left(\dfrac{1}{2}\right)^3\right]^2\)

\(=>\left(\dfrac{1}{2}\right)^n=\left(\dfrac{1}{2}\right)^6\)

\(\Rightarrow n=6\)

a) Gọi H là giao điểm đường trung trực của EF và EF

Xét Δ KEF có :

KH là đường trung trực của EF

⇒ KH vừa là đường cao, trung tuyến của Δ KEF

⇒ Δ KEF là tam giác cân tại K

b) Xét Δ vuông DEF có :

\(\widehat{DEF}+\widehat{DFE}=90^o\)

\(\Rightarrow\widehat{DEF}=90^o-\widehat{DFE}\)

\(\Rightarrow\widehat{DEF}=90^o-40^o\)

\(\Rightarrow\widehat{DEF}=50^o\)

mà \(\widehat{DEK}+\widehat{KEF}=\widehat{DEF}\)

     \(\widehat{KEF}=\widehat{DFE}=40^o\) (Δ KEF là tam giác cân tại K)

\(\Rightarrow\widehat{DEK}=\widehat{DEF}-\widehat{KEF}=50^o-40^o=10^o\)

Quãng đường đó thực tế dài là :

\(30.120000=3600000=36.10^5=3,6.10^6\left(cm\right)\)

29) 2.x⁴ = 2.3⁴

x⁴ = 3⁴

x = 3 hoặc x = -3

30) 5.x³ = 5.4³

x³ = 4³

x = 4

31) 7/5 . x⁵ = 7/5 . 4⁵

x⁵ = 4⁵

x = 4

32) 3/4 . x¹² = 3/4 . 5¹²

x¹² = 5¹²

x = 5 hoặc x = -5

33) 9.x⁶ = 9.7⁶

x⁶ = 7⁶

x = 7 hoặc x = -7

34) x⁹ = 2.2⁸

x⁹ = 2⁹

x = 2

35) x¹² = 5⁴.5⁸

x¹² = 5¹²

x = 5 hoặc x = -5

36) 5.3ˣ = 7.3⁵ - 2.3⁵

5.3ˣ = 3⁵.(7 - 2)

5.3ˣ = 5.3⁵

3ˣ = 3⁵

x = 5

37) 9.x⁶ = 6.5⁶ + 3.5⁶

9.x⁶ = (6 + 3).5⁶

9.x⁶ = 9.5⁶

x⁶ = 5⁶

x = 5 hoặc x = -5

38) x¹⁰ = 4.4¹⁰ - 3.4¹⁰

x¹⁰ = (4 - 3).4¹⁰

x¹⁰ = 4¹⁰

x = 4 hoặc x = -4

39) 7.x⁷ = 5.3⁷ + 2.3⁷

7.x⁷ = (5 + 2).3⁷

= 7.x⁷ = 7.3⁷

x⁷ = 3⁷

x = 3

40) 7.x⁸ = 2⁹ + 5.2⁸

7.x⁸ = (2 + 5).2⁸

7.x⁸ = 7.2⁸

x⁸ = 2⁸

x = 2 hoặc x = -2

41) 5.x¹⁰ = 8.3¹⁰ - 3.3¹⁰

5.x¹⁰ = (8 - 3).3¹⁰

5.x¹⁰ = 5.3¹⁰

x¹⁰ = 3¹⁰

x = 3 hoặc x = -3

42) 5.(x + 6)⁵ = 2.3⁵ + 3.3⁵

5.(x + 6)⁵ = (2 + 3).3⁵

5.(x + 6)⁵ = 5.3⁵

(x + 6)⁵ = 3⁵

x + 6 = 3

x = 3 - 6

x = -3

43) (2x + 1)⁸ = 5.9⁷ + 4.9⁷

(2x + 1)⁸ = (5 + 4).9⁷

(2x + 1)⁸ = 9.9⁷

(2x + 1)⁸ = 9⁸

2x + 1 = 9 hoặc 2x + 1 = -9

*) 2x + 1 = 9

2x = 8

x = 4

*) 2x + 1 = -9

2x = -10

x = -5

Vậy x = -5; x = 4

44) (x + 1)³ + 4.(x + 1)³ = 5.3⁶

5(x + 1)³ = 5.(3²)³

(x + 1)³ = 9³

x + 1 = 9

x = 8

45) 1/5 . (x + 2)² + 1/3 . (2x - 2)³ = 1/5 . (x + 2)² + 1/3 . 2³

1/3 . (2x  - 2)³ = 1/3 . 2³

(2x - 2)³ = 2³

2x - 2 = 2

2x = 4

x = 2

46) 2/3 . (2x + 4)² - 1/3 . (x + 1)² = -1/3 . (2x + 4)² + 2/3 (x + 1)²

2/3 (2x + 4)² + 1/3 (2x + 4)² - 1/3 (x + 1)² - 2/3 (x + 1)² = 0

(2x + 4)² - (x + 1)² = 0

(2x + 4)² = (x + 1)²

2x + 4 = x + 1 hoặc 2x + 4 = -x - 1

*) 2x + 4 = x + 1

2x - x = 1 - 4

x = -3

*) 2x + 4 = -x - 1

2x + x = -1 - 4

3x = -5

x = -5/3

Vậy x = -3; x = -5/3

a) \(x^2-\dfrac{4}{3}x=0\)

\(\Rightarrow x\left(x-\dfrac{4}{3}\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\x-\dfrac{4}{3}=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{4}{3}\end{matrix}\right.\)

b) \(\left|3x+7\right|=5\)

\(\Rightarrow\left[{}\begin{matrix}3x+7=5\\3x+7=-5\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}3x=-2\\3x=-12\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=-\dfrac{2}{3}\\x=-4\end{matrix}\right.\)

c) \(2\left|7-2x\right|=6\)

\(\Rightarrow\left|7-2x\right|=3\)

\(\Rightarrow\left[{}\begin{matrix}7-2x=3\\7-2x=-3\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}2x=4\\2x=10\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=2\\x=5\end{matrix}\right.\)

d) \(5\left(x+2\right)+2\left(x-1\right)=10\)

\(\Rightarrow5x+10+2x-2=10\)

\(\Rightarrow5x+2x+8=10\)

\(\Rightarrow7x=2\)

\(\Rightarrow x=\dfrac{2}{7}\)

e) \(6\left(x-3\right)-4\left(x+2\right)=15\)

\(\Rightarrow6x-18-4x-8=15\)

\(\Rightarrow6x-4x-18-8=15\)

\(\Rightarrow2x-26=15\)

\(\Rightarrow2x=41\)

\(\Rightarrow x=\dfrac{41}{2}\)

g) \(\left(12-2x\right)^2=\dfrac{4}{9}\)

\(\Rightarrow\left[{}\begin{matrix}12-2x=\dfrac{2}{3}\\12-2x=-\dfrac{2}{3}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}12x=12-\dfrac{2}{3}\\12x=12+\dfrac{2}{3}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}12x=\dfrac{34}{3}\\12x=\dfrac{38}{3}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{34}{3}.\dfrac{1}{12}\\x=\dfrac{38}{3}.\dfrac{1}{12}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{17}{18}\\x=\dfrac{19}{18}\end{matrix}\right.\)

\(\dfrac{x}{2}=\dfrac{4}{5}=4.2:5=1,6\)

2x​=54​=4.2:5=1,6

\(0,6^2.x=0,36^3\)

\(0,36.x=0,36^3\)

\(\Rightarrow x=0,36^3:0,36\)

\(\Rightarrow x=0,36^{3-1}\)

\(\Rightarrow x=0,36^2\)

A = \(\dfrac{3n+1}{2n+3}\) (n \(\ne\) - \(\dfrac{3}{2}\))

A \(\in\) Z ⇔ 3n + 1 ⋮ 2n + 3

             6n + 2 ⋮ 2n + 3

         6n + 9 - 7 ⋮ 2n + 3

    3.(2n + 3) - 7 ⋮ 2n + 3

                      7 ⋮ 2n + 3 ⇒ 2n + 3 \(\in\) Ư(7) = { -7; -1; 1; 7}

Lập bảng ta có: 

Vậy các số nguyên n thỏa mãn đề bài là:

n \(\in\) { -5; -2; -1; 2}

\(A=\dfrac{3n+1}{2n+3}\inℤ\) \(\left(n\ne-\dfrac{3}{2}\right)\)

\(\Rightarrow3n+1⋮2n+3\)

\(\Rightarrow2\left(3n+1\right)-3\left(2n+3\right)⋮2n+3\)

\(\Rightarrow6n+2-6n-9⋮2n+3\)

\(\Rightarrow-7⋮2n+3\)

\(\Rightarrow2n+3\in\left\{-1;1;-7;7\right\}\)

\(\Rightarrow n\in\left\{-2;-1;-5;2\right\}\)

\(\dfrac{2}{x}\) + \(\dfrac{1}{y}\) = \(\dfrac{1}{6}\) (\(x;y\) \(\in\) N*)

\(\dfrac{2}{x}\)         = \(\dfrac{1}{6}\) - \(\dfrac{1}{y}\)

\(\dfrac{2}{x}\)         = \(\dfrac{y-6}{6y}\)

\(x\)          = 2: \(\dfrac{y-6}{6y}\)

\(x\)         = \(\dfrac{12y}{y-6}\)

Vì \(x\); y \(\in\) N* nên 12\(y\) ⋮ y - 6 ( và y > 6)

12y ⋮ y - 6 ⇔ 12y - 72 + 72 ⋮ y - 6 ⇔ 12.(y-6) + 72 ⋮ y - 6 ⇔ 72⋮ y - 6  72 = 2³.3² 

Ư(72) = { 1; 2; 3; 4; 6; 8; 9; 12; 18; 24; 36; 72}

Lập bảng ta có:

Theo bảng trên ta có các cặp số tự nhên \(x\); y thỏa mãn đề bài lần lượt là:

(\(x\);y) =(84;7); (48;8); (36;9); (30;10);(34;12); (21;14); (20;15);(18;18);

(16;24); (15; 30); (14;42);(13;78)

\(\dfrac{2}{x}+\dfrac{1}{y}=\dfrac{1}{6}\left(x;y\inℕ^∗\right)\)

\(\Leftrightarrow\dfrac{2y+x}{xy}=\dfrac{1}{6}\)

\(\Leftrightarrow6\left(2y+x\right)=xy\)

\(\Leftrightarrow12y+6x=xy\)

\(\Leftrightarrow12y-xy+6x=0\)

\(\Leftrightarrow y\left(12-x\right)+6x-72+72=0\)

\(\Leftrightarrow-y\left(x-12\right)+6\left(x-12\right)=-72\)

\(\Leftrightarrow\left(x-12\right)\left(6-y\right)=-72\)

\(\Leftrightarrow\left(x-12\right);\left(6-y\right)\in\left\{-1;1;-2;2;-3;3;-4;4;-8;8;-9;9;-18;18;-24;24;-36;36;-72;72\right\}\)

Lập bảng sẽ ra \(\left(x;y\inℕ^∗\right)\) cần tìm...