\(6^2.2^2.5:\left[3.12-\left(2x-6\right)\right]=2^3.5\)

\(\Rightarrow36.4.5:\left[36-\left(2x-6\right)\right]=8.5\)

\(\Rightarrow720:\left[36-\left(2x-6\right)\right]=40\)

\(\Rightarrow36-\left(2x-6\right)=720:40=18\)

\(\Rightarrow2x-6=36-18=18\)

\(\Rightarrow2x=18+6=24\)

\(\Rightarrow x=24:2=12\)

`#3107`

b)

`2.3^x = 162`

`\Rightarrow 3^x = 162 \div 2`

`\Rightarrow 3^x = 81`

`\Rightarrow 3^x = 3^4`

`\Rightarrow x = 4`

Vậy, `x = 4`

c)

`(2x - 15)^5 = (2 - 15)^3`

\(\Rightarrow \)`(2x - 15)^5 - (2x - 15)^3 = 0`

\(\Rightarrow \)`(2x - 15)^3 . [ (2x - 15)^2 - 1] = 0`

\(\Rightarrow\left[{}\begin{matrix}\left(2x-15\right)^3=0\\\left(2x-15\right)^2-1=0\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}2x-15=0\\\left(2x-15\right)^2=1\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}2x=15\\\left(2x-15\right)^2=\left(\pm1\right)^2\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=\dfrac{15}{2}\\2x-15=1\\2x-15=-1\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=\dfrac{15}{2}\\2x=16\\2x=-14\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=\dfrac{15}{2}\\x=8\\x=-7\end{matrix}\right.\)

Vậy, `x \in`\(\left\{-7;8;\dfrac{15}{2}\right\}.\)

`d)`

\(3^{x+2}-5.3^x=?\) Bạn ghi tiếp đề nhé!

`e)`

\(7\cdot4^{x-1}+4^{x-1}=23?\)

\(4^{x-1}\cdot\left(7+1\right)=23\\ \Rightarrow4^{x-1}\cdot8=23\\ \Rightarrow4^{x-1}=\dfrac{23}{8}\)

Bạn xem lại đề!

`f)`

\(2\cdot2^{2x}+4^3\cdot4^x=1056\)

\(\Rightarrow2\cdot2^{2x}+\left(2^2\right)^3\cdot\left(2^2\right)^x=1056\\ \Rightarrow2\cdot2^{2x}+2^6\cdot2^{2x}=1056\\ \Rightarrow2^{2x}\cdot\left(2+2^6\right)=1056\\ \Rightarrow2^{2x}\cdot66=1056\\ \Rightarrow2^{2x}=1056\div66\\ \Rightarrow2^{2x}=16\\ \Rightarrow2^{2x}=2^4\\ \Rightarrow2x=4\\ \Rightarrow x=2\)

Vậy, `x = 2`

_____

\(10 -{[(x \div 3+17) \div 10+3.2^4] \div 10}=5\)

\(\Rightarrow\left[\left(x\div3+17\right)\div10+48\right]\div10=10-5\)

\(\Rightarrow\left[\left(x\div3+17\right)\div10+48\right]\div10=5\)

\(\Rightarrow\left(x\div3+17\right)\div10+48=50\)

\(\Rightarrow\left(x\div3+17\right)\div10=2\)

\(\Rightarrow x\div3+17=20\)

\(\Rightarrow x\div3=3\\ \Rightarrow x=9\)

Vậy, `x = 9.`

A = 2.2² + 3.2³ + 4.2⁴ + ... + n.2ⁿ 

2.A = 2.2³ + 3.2⁴ + 4.2⁵ + ...+ n.2ⁿ⁺¹

=> A - 2.A = 2.2² + (3.2³ - 2.2³)  + (4.2⁴ - 3.2⁴) + ...+ (n - n + 1).2ⁿ - n.2ⁿ⁺¹

=> A = 2.2² + 2³ + 2⁴ + ..+ 2ⁿ - n.2^{n+ 1}  = 2² + (2² + 2³ + ....+ 2^{n+ 1}) - (n+1).2ⁿ⁺¹

=> A =  - 2² -  (2² + 2³ + ....+ 2^{n+ 1}) + (n+1).2ⁿ⁺¹

Tính B = 2² + 2³ + ....+ 2^{n+ 1} => 2.B =  2³ + ....+ 2^{n+ 1} + 2ⁿ⁺² => 2B - B = 2ⁿ⁺² - 2² => B = 2ⁿ⁺² - 2²

Vậy A = 2² - 2ⁿ⁺² + 2² + (n+1).2ⁿ⁺¹ = (n+1).2ⁿ⁺¹ - 2^{n+ 2} = 2ⁿ⁺¹.(n + 1 - 2) = (n-1).2ⁿ⁺¹ = 2(n-1).2ⁿ

Theo bài cho  A = 2(n-1).2ⁿ = 2ⁿ⁺¹⁰ => 2(n - 1) = 2¹⁰ => n - 1 = 2⁹  = 512 => n = 513

Vậy.............

n= 513, tui chỉ biết đáp án nhưng không biết cách làm

đặt A=2+2^2+2^3+...+2^n

     2A=2^2+2^3+2^4+...+2^n+1 (1)

  2A-A=2\(^{n+1}\)-2

     A=2\(^{n+1}\)-2  (2)

từ (1)(2) =>2 + 2\(^2\)+2\(^3\)+...+2\(^n\)=2\(^{n+1}\)-2

                      2\(^2\)+2\(^3\)+...+2\(^n\)=2\(^{n-1}\)-2\(^2\)

                              ..............................

                                             2\(^n\)=2\(^{n-1}\)-2\(^n\)

cộng vế với vế ta có 

 2+2.2\(^2\)+3.2\(^3\)+...+n.2\(^n\)= n.2\(^{n+1}\)- (2+2\(^2\)+2\(^3\)+...+2\(^n\))

2+(2.2\(^2\)+3.2\(^3\)+...+n.2\(^n\)=n.2\(^{n+1}\)- A

     2+2\(^{n+10}\)=n.2\(^{n+1}\)-2\(^{n+1}\)+2

            2\(^{n+10}\)=2\(^{n+1}\).(n-1)

             2\(^{n+1}\). 2\(^9\)=2\(^{n+1}\).(n-1)

=>n-1=2\(^9\)

=>n=2^9+1=513

vậy n=513

a) (2x - 17 )⁵ = (2x -17 )⁵

=> Với mọi x \(\in\)N thì : ( 2x - 17 )⁵ = (2x -17 )⁵

Vậy :...

b) 

=> Với x = 0 hoặc x = 1 thì  5^x = 3^x

Vậy :...

a) Ta có:  ( x + 1 ) 4 = ( 2 x ) 4 nên x +1 = 2x. Do đó x = 1.

b) Ta có:  ( 2 x - l ) 5 = x 5 nên 2x - l = x. Do đó x = l.

a,(2x+1)(y-3)=12

2x+1 và y-3 Ư(12)=

=>x=0,y=15

c) Ta có: \(36^{25}=\left(6^2\right)^{25}=6^{50}\)

\(25^{36}=\left(5^2\right)^{36}=5^{72}\)

Ta có: \(6^{50}=\left(6^5\right)^{10}=7776^{10}\)

mà \(5^{70}=\left(5^7\right)^{10}=78125^{10}\)

nên \(6^{50}< 5^{70}\)

mà \(5^{70}< 5^{72}\)

nên \(6^{50}< 5^{72}\)

hay \(36^{25}< 25^{36}\)

a/

Với $x,y$ là số tự nhiên $2x+1, y-3$ là số nguyên. Mà $(2x+1)(y-3)=12$ nên $2x+1$ là ước của 12. 

$2x+1>0, 2x+1$ lẻ nên $2x+1\in \left\{1;3\right\}$

Nếu $2x+1=1\Rightarrow y-3=12$

$\Rightarrow x=0; y=15$

Nếu $2x+1=3\Rightarrow y-3=4$

$\Rightarrow x=1; y=7$ 

Vậy...........

b/

$2^x+2^{x+1}+2^{x+2}+...+2^{x+2015}=2^{2019}-8$

$2^x(1+2+2^2+2^3+...+2^{2015})=2^{2019}-8(1)$
$2^x(2+2^2+2^3+2^4+...+2^{2016})=2^{2020}-16(2)$ (nhân 2 vế với 2)

Lấy (2) trừ (1) theo vế thì:

$2^x(2^{2016}-1)=2^{2020}-2^{2019}-8$

$2^x(2^{2016}-1)=2^{2019}(2-1)-8=2^{2019}-8$

$2^x(2^{2016}-1)=2^3(2^{2016}-1)$

$\Rightarrow 2^x=2^3$

$\Rightarrow x=3$

c/

$25^{36}=(5^2)^{36}=5^{72}$

$36^{25}=(6^2)^{25}=6^{50}=(6^5)^{10}< (5^7)^{10}=5^{70}< 5^{72}$

$\Rightarrow 25^{36}> 36^{25}$