Discussion

Explanation:

Given, $\left(\sqrt[-\mathrm{pq}]{{\mathrm{a}}^{\mathrm{r}}}\right)\times \left(\sqrt[-\mathrm{qr}]{{\mathrm{a}}^{\mathrm{p}}}\right)\times \left(\sqrt[-\mathrm{pr}]{{\mathrm{a}}^{\mathrm{q}}}\right)$

Let’s consider $\sqrt[-\mathrm{pq}]{{\mathrm{a}}^{\mathrm{r}}}$.

⇒ $\sqrt[-\mathrm{pq}]{{\mathrm{a}}^{\mathrm{r}}}$ = ${\left({\mathrm{a}}^{\mathrm{r}}\right)}^{-\frac{1}{pq}}$ = ${\mathrm{a}}^{-\frac{\mathrm{r}}{\mathrm{pq}}}$   …(1)

Similarly, 

$\sqrt[-\mathrm{qr}]{{\mathrm{a}}^p}$ = ${\mathrm{a}}^{-\frac{\mathrm{p}}{\mathrm{qr}}}$   …(2)

$\sqrt[-\mathrm{rp}]{{\mathrm{a}}^q}$ = ${\mathrm{a}}^{-\frac{q}{\mathrm{rp}}}$   …(3)

Multiplying all three equation,

$\left(\sqrt[-\mathrm{pq}]{{\mathrm{a}}^{\mathrm{r}}}\right)\times \left(\sqrt[-\mathrm{qr}]{{\mathrm{a}}^{\mathrm{p}}}\right)\times \left(\sqrt[-\mathrm{pr}]{{\mathrm{a}}^{\mathrm{q}}}\right)$ = ${\mathrm{a}}^{-\frac{\mathrm{r}}{\mathrm{pq}}}\times {\mathrm{a}}^{-\frac{\mathrm{p}}{\mathrm{qr}}}\times {\mathrm{a}}^{-\frac{q}{\mathrm{rp}}}$

= ${\mathrm{a}}^{-\frac{\mathrm{r}}{\mathrm{pq}}}\times {\mathrm{a}}^{-\frac{\mathrm{p}}{\mathrm{qr}}}\times {\mathrm{a}}^{-\frac{q}{\mathrm{rp}}}$

= ${\mathrm{a}}^{-\left(\frac{{\mathrm{r}}^2+p^2+q^2}{\mathrm{pqr}}\right)}$

= ${\mathrm{a}}^{-\left(\frac{1}{\mathrm{pqr}}\right)}$ [p² + q² + r² = 1]

Hence, option (d).

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