Question

সরল কর: $\sqrt{x^{-1}y}\cdot\sqrt{y^{-1}z}\cdot\sqrt{z^{-1}x}$ $\left(x>0,y>0,z>0\right)$

Answer 1

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

$=\sqrt{x^{-1}y \cdot y^{-1}z \cdot z^{-1}x}$

$=\sqrt{x^{-1+1}\cdot y^{-1+1}\cdot z^{-1+1}}$

$=\sqrt{x^0\cdot y^0\cdot z^0}$

$=\sqrt{1\cdot1\cdot1}$

$=\sqrt1$

$=1$ [Answer]

Answer 2

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

$=\sqrt{\frac{1}{x}y}\cdot\sqrt{\frac{1}{y}z}\cdot\sqrt{\frac{1}{z}x}$

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

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

$=\sqrt1$

$=1$ [Answer]