Question

[Võ Bảo Chung]

Tìm x,y,z biết:

x căn (yz)=8 ; y căn (xz) =2; z căn (xy)=1

Answer 1

[Hoàng Thị Lan Hương]

ĐK \(x;y;z>0\)

Đặt \(x\sqrt{yz}=\left(1\right);y\sqrt{xz}=\left(2\right);z\sqrt{xy}=\left(3\right)\)

Lấy \(\frac{\left(1\right)}{\left(2\right)}\)ta có \(\frac{x\sqrt{yz}}{y\sqrt{xz}}=\frac{x}{y}.\sqrt{\frac{y}{x}}=\frac{8}{2}=4\Rightarrow\frac{x^2}{y^2}.\frac{y}{x}=16\Rightarrow\frac{x}{y}=16\)\(\Rightarrow x=16y\)

Tương tự ta có \(\frac{y\sqrt{xz}}{z\sqrt{xy}}=2\Rightarrow\frac{y}{z}=4\Rightarrow z=\frac{y}{4}\)

Thay x;z vào (2) ta có \(y\sqrt{xz}=y\sqrt{16y.\frac{y}{4}}=2\Rightarrow y^2=1\Rightarrow\orbr{\begin{cases}y=1\\y=-1\left(l\right)\end{cases}\Rightarrow y=1}\)

\(\Rightarrow x=16;z=\frac{1}{4}\)

Vậy \(x=16;y=1;z=\frac{1}{4}\)