\(\frac{x\sqrt{y}-y\sqrt{x}}{\sqrt{x}-\sqrt{y}}\)

\(=\frac{\left(x\sqrt{y}-y\sqrt{x}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)

\(=\frac{\sqrt{xy}.x+xy-xy-\sqrt{xy}.y}{x-y}\)

\(=\frac{\sqrt{xy}.x-\sqrt{xy}.y}{x-y}\)

\(\frac{=\sqrt{xy}.\left(x-y\right)}{x-y}\)

\(=\sqrt{xy}\)

a, \(\sqrt{16x}=8\)ĐK : x >= 0 

\(\Leftrightarrow4\sqrt{x}=8\Leftrightarrow\sqrt{x}=2\Leftrightarrow x=4\)

c, \(\sqrt{9\left(x-1\right)}=21\Leftrightarrow3\sqrt{x-1}=21\Leftrightarrow\sqrt{x-1}=7\)ĐK : x >= 1 

\(\Leftrightarrow x-1=49\Leftrightarrow x=50\)

t học lớp 8 bạn ei :)

ĐKXĐ : x \(\ge2\)

Ta có \(\sqrt{x-2}-\sqrt{x+2}=2\sqrt{x^2-4}-2x+2\)

\(\Leftrightarrow\sqrt{x-2}-\sqrt{x+2}=2\sqrt{\left(x-2\right)\left(x+2\right)}-\left(x-2\right)-\left(x+2\right)+2\)

<=> \(\sqrt{x-2}-\sqrt{x+2}=-\left(\sqrt{x-2}-\sqrt{x+2}\right)^2+2\)

Đặt \(\sqrt{x-2}-\sqrt{x+2}=y\)

=> y = -y² + 2 

<=> y² - y - 2 = 0

<=> (y + 1)(y - 2) = 0

<=> \(\orbr{\begin{cases}y=-1\\y=2\end{cases}}\)

Khi y = -1

<=> \(\sqrt{x-2}-\sqrt{x+2}=-1\)

=> \(\left(\sqrt{x-2}-\sqrt{x+2}\right)^2=1\)

<=> \(\left(x-2\right)+\left(x+2\right)-2\sqrt{\left(x-2\right)\left(x+2\right)}=1\)

<=> \(2x-1=2\sqrt{\left(x-2\right)\left(x+2\right)}\)

=> 4x² - 4x + 1 = 2(x - 2)(x + 2) 

<=> 4x² - 4x + 1 = 2x² - 8 

<=> 2x² - 4x + 9 = 0 (vô lý) => TH1 loại

Khi y = 2 =>\(\sqrt{x-2}-\sqrt{x+2}=2\)

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

<=> \(2x-2\sqrt{\left(x-2\right)\left(x+2\right)}=4\)

<=> \(2x-4=2\sqrt{\left(x-2\right)\left(x+2\right)}\)

=> (2x - 4)² = 4(x - 2)(x + 2)

<=> 4(x - 2)² = 4(x - 2)(x + 2) 

<=> -16(x - 2) = 0

<=> x = 2 (tm) 

Vậy x = 2

\(A=\frac{10a^2+10b^2+c^2}{ab+bc+ca}=\frac{8a^2+\frac{c^2}{2}+8b^2+\frac{c^2}{2}+2a^2+2b^2}{ab+bc+ca}\)

\(\ge\frac{2\sqrt{8a^2.\frac{c^2}{2}}+2\sqrt{8b^2.\frac{c^2}{2}}+4\sqrt{a^2b^2}}{ab+bc+ca}=\frac{4\left(ab+bc+ca\right)}{ab+bc+ca}=4\)

Dấu \(=\)khi \(a=b=\frac{c}{4}\).

Bạn tham khảo nhé: áp dụng bđt côsi cho 2 số dương

2a²+2b²>=4ab;8a²+c²/2>=4ac;8b²+c²/2>=4ac nên A>=4

dấu bằng xảy ra khi 4a=4b=c

\(D=\sqrt{9-\sqrt{87}}\sqrt{9+\sqrt{87}}=\sqrt{81-87}\)

đề sai ko bạn vì \(\sqrt{a}\)xảy ra khi a >= 0 mà -6 < 0 bạn nhé