Question

If x, y, z are positive real numbers, show that $\sqrt{x^{-1}y}\times \sqrt{y^{-1}z}\times \sqrt{z^{-1}x}=1$.

Answer 1

$\sqrt{x^{-1}y}\times \sqrt{y^{-1}z}\times \sqrt{z^{-1}x}$

$=\sqrt{\frac{y}{x}}\times \sqrt{\frac{z}{y}}\times \sqrt{\frac{x}{z}}$ [$\because a^{-1}=\frac{1}{a}$]

$=\frac{\sqrt{y}}{\sqrt{x}}\times \frac{\sqrt{z}}{\sqrt{y}}\times \frac{\sqrt{x}}{\sqrt{z}}$

$=1$