Bài làm :

Ta có:

\(x^2-y^2-z^2=0\)

\(\Leftrightarrow16x^2-16y^2-16z^2=0\)

\(\Leftrightarrow25x^2-9x^2+9y^2-25y^2-16z^2+30xy-30xy=0\)

\(\Leftrightarrow\left[\left(25x^2-30xy+9y^2\right)-16z^2\right]-\left(9x^2-30xy+25y^2\right)=0\)

\(\Leftrightarrow\left(5x-3y\right)^2-16z^2=\left(3x-5y\right)^2\)

\(\Leftrightarrow\left(5x-3y-4z\right)\left(5x-3y+4z\right)=\left(3x-5y\right)^2\)

=> Điều phải chứng minh

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

Dạng Chính Xác:

3√3+63√2+6√432

Dạng Thập Phân:

15.50513799…

a) \(A=\sqrt{19+8\sqrt{3}}-\sqrt{4+2\sqrt{3}}\)

\(A=\sqrt{16+8\sqrt{3}+3}-\sqrt{3+2\sqrt{3}+1}\)

\(A=\sqrt{\left(4+\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{3}+1\right)^2}\)

\(A=4+\sqrt{3}-\sqrt{3}-1=3\)

b) \(B=\sqrt{27+10\sqrt{2}}-\sqrt{18+8\sqrt{2}}\)

\(B=\sqrt{25+10\sqrt{2}+2}-\sqrt{16+8\sqrt{2}+2}\)

\(A=\sqrt{\left(5+\sqrt{2}\right)^2}-\sqrt{\left(4+\sqrt{2}\right)^2}\)

\(A=5+\sqrt{2}-4-\sqrt{2}=1\)

\(A=\sqrt{19+8\sqrt{3}}-\sqrt{4+2\sqrt{3}}\)

\(=\sqrt{3+8\sqrt{3}+16}-\sqrt{3+2\sqrt{3}+1}\)

\(=\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot4+4^2}-\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}+1^2}\)

\(=\sqrt{\left(\sqrt{3}+4\right)^2}-\sqrt{\left(\sqrt{3}+1\right)^2}\)

\(=\left|\sqrt{3}+4\right|-\left|\sqrt{3}+1\right|\)

\(=\sqrt{3}+4-\left(\sqrt{3}+1\right)\)

\(=\sqrt{3}+4-\sqrt{3}-1=3\)

\(B=\sqrt{27+10\sqrt{2}}-\sqrt{18+8\sqrt{2}}\)

\(=\sqrt{2+10\sqrt{2}+25}-\sqrt{2+8\sqrt{2}+16}\)

\(=\sqrt{\left(\sqrt{2}\right)^2+2\cdot\sqrt{2}\cdot5+5^2}-\sqrt{\left(\sqrt{2}\right)^2+2\cdot\sqrt{2}\cdot4+4^2}\)

\(=\sqrt{\left(\sqrt{2}+5\right)^2}-\sqrt{\left(\sqrt{2}+4\right)^2}\)

\(=\left|\sqrt{2}+5\right|-\left|\sqrt{2}+4\right|\)

\(=\sqrt{2}+5-\left(\sqrt{2}+4\right)\)

\(=\sqrt{2}+5-\sqrt{2}-4=1\)

                  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