Bài làm : 

Ta có :

\(\left(8x-1\right)^{2n+1}=5^{2n+1}\)

\(\Leftrightarrow8x-1=5\)

\(\Leftrightarrow8x=5+1\)

\(\Leftrightarrow8x=6\)

\(\Leftrightarrow x=\frac{6}{8}=\frac{3}{4}\)

Chúc bạn học tốt !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Vì n là số tự nhiên => 2n + 1 là số lẻ

Khi đó (8x - 1)^{2n + 1} = 5^{2n + 1}

<=> 8x  - 1 = 5

=> 8x = 6

=> x = 0,75

Vậy x = 0,75

                      Bài làm : 

Ta có:

 \(P=12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)

\(P=\frac{24\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)}{2}\)

\(P=\frac{\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)}{2}\)

\(P=\frac{\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)}{2}\)

\(P=\frac{\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)}{2}\)

\(P=\frac{\left(5^{16}-1\right)\left(5^{16}+1\right)}{2}\)

\(P=\frac{5^{32}-1}{2}\)

\(\text{Vậy : }P=\frac{5^{32}-1}{2}\)

Chúc bạn học tốt !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Bài làm:

Đặt \(A=12\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)

=> \(2A=24\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)

<=> \(2A=\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)

<=> \(2A=\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)

<=> \(2A=\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)\)

<=> \(2A=\left(5^{16}-1\right)\left(5^{16}+1\right)\)

<=> \(2A=5^{32}-1\)

=> \(A=\frac{5^{32}-1}{2}\)

\(a\)

\(\sqrt{11}+\sqrt{19}\)

\(=\)\(\sqrt{11+19}\)

\(=\)\(\sqrt{30}\)

\(=\)\(5,47\)

\(\sqrt{47}\)

\(=6,85\)

\(5,47\)\(< \)\(6,85\)

\(=>\)\(\sqrt{11}+\sqrt{19}\)\(< \)\(\sqrt{47}\)

\(b\)

\(\sqrt{7}+\sqrt{26}+1\)

\(=\)\(\sqrt{7+26}+1\)

\(=\)\(\sqrt{33}+1\)

\(=\)\(5,74+1\)

\(=\)\(6,74\)

\(\sqrt{63}\)

\(=\)\(7,93\)

\(6,74\)\(< \)\(7,93\)

\(=>\)\(\sqrt{7}+\sqrt{26}+1\)\(< \)\(\sqrt{63}\)

Học tốt!!!

\(\left(x-\frac{2}{9}\right)^3=\left(\left(\frac{2}{3}\right)^2\right)^3\)        

\(\left(x-\frac{2}{9}\right)^3=\left(\frac{4}{9}\right)^3\)       

\(x-\frac{2}{9}=\frac{4}{9}\)     

\(x=\frac{2}{3}\)

Bài làm:

Ta có: \(\left(x-\frac{2}{9}\right)^3=\left(\frac{2}{3}\right)^6\)

\(\Leftrightarrow\left(x-\frac{2}{9}\right)^3=\left(\frac{4}{9}\right)^3\)

\(\Leftrightarrow x-\frac{2}{9}=\frac{4}{9}\)

\(\Rightarrow x=\frac{6}{9}=\frac{2}{3}\)

( 5x + 1 )² = 36/49

<=> ( 5x + 1 )² = ( ±6/7 )²

<=> 5x + 1 = 6/7 hoặc 5x + 1 = -6/7

<=> x = -1/35 hoặc x = -13/35

\(\left(5x+1\right)^2=\frac{36}{49}\)

\(\Rightarrow\orbr{\begin{cases}5x+1=\frac{6}{7}\\5x+1=\frac{-6}{7}\end{cases}}\)\(\)

\(\Rightarrow\orbr{\begin{cases}5x=\frac{-1}{7}\\5x=\frac{-13}{7}\end{cases}}\)

\(\Rightarrow\orbr{\begin{cases}x=\frac{-1}{35}\\x=\frac{-13}{35}\end{cases}}\)

Bài làm:

Ta có: \(\left(\frac{2}{5}\right)^x>\left(\frac{5}{2}\right)^{-3}\cdot\left(-\frac{2}{5}\right)^2\)

\(\Leftrightarrow\left(\frac{2}{5}\right)^x>\left(\frac{2}{5}\right)^3.\left(\frac{2}{5}\right)^2\)

\(\Leftrightarrow\left(\frac{2}{5}\right)^x>\left(\frac{2}{5}\right)^5\)

\(\Rightarrow x>5\)

=> \(x\in N=\left\{x\in N;x>5\right\}\)

Trả lời :

27 < 81³ : 3^x < 243

=> 3³ < 3⁴ : 3^x < 3⁵

=> 3³ < 3^{4 - x} < 3⁵

=> 3 < 4 - x < 5

=> 4 - x = 4

Mà x \(\inℕ\)=> x = 0

Ta có : a.bcd.abc = abcabc

=> a.bcd.abc = abc.1001

=> a.bcd = 1001 ( Vì \(abc\ne0\))

Vì a ; bcd đều là số tự nhiên mà a là số có 1 chữ số (\(a\ne0\))

Phân tích ra các thừa số ta đươc : 1001 = 7 . 13 .11

Dễ dàng nhận thấy a = 7 

và bcd = 13.11

<=> bcd = 143

Vậy a = 7 ; b = 1 ; c = 4 ; d = 3 

Ta có: \(8< 2^x< 2^9.2^{-5}\)

\(\Leftrightarrow2^3< 2^x< 2^4\)

Mà x là số tự nhiên

=> Không tồn tại x thỏa mãn

Bài làm:

Ta có: \(80^2-75^2+70^2-65^2+60^2-...+10^2-5^2\)

\(=\left(80-75\right)\left(80+75\right)+\left(70-65\right)\left(70+65\right)+...+\left(10-5\right)\left(10+5\right)\)

\(=5.\left(80+75\right)+5.\left(70+65\right)+...+5.\left(10+5\right)\)

\(=5.\left(80+75+70+65+...+10+5\right)\)

\(=5\cdot\frac{\left(5+80\right)\cdot\left[\left(80-5\right)\div5+1\right]}{2}\)

\(=5\cdot\frac{85\cdot16}{2}=3400\)