The value of
Question:

The value of $\sqrt{p^{-1} q} \cdot \sqrt{q^{-1} r} \cdot \sqrt{r^{-1} p}$ is

(a) $-1$

(b) 0

(C) 1

(d) 2

Solution:

$\sqrt{p^{-1} q} \cdot \sqrt{q^{-1} r} \cdot \sqrt{r^{-1} p}=\left(p^{-1} q\right)^{\frac{1}{2}} \cdot\left(q^{-1} r\right)^{\frac{1}{2}} \cdot\left(r^{-1} p\right)^{\frac{1}{2}}$

$=\left(p^{-\frac{1}{2}} q^{\frac{1}{2}}\right) \cdot\left(q^{-\frac{1}{2}} r^{\frac{1}{2}}\right) \cdot\left(r^{-\frac{1}{2}} p^{\frac{1}{2}}\right)$

$=p^{-\frac{1}{2}+\frac{1}{2}} q^{\frac{1}{2}-\frac{1}{2}} r^{\frac{1}{2}-\frac{1}{2}}$

$=p^{0} q^{0} r^{0}$

$=1$

$\therefore$ The value of $\sqrt{p^{-1} q} \cdot \sqrt{q^{-1} r} \cdot \sqrt{r^{-1} p}$ is equal to 1 .

Hence, the correct option is (c).