Question

Simplify: $\sqrt[qr]{\frac{x^q}{x^r}}\times \sqrt[rp]{\frac{x^r}{x^p}}\times \sqrt[pq]{\frac{x^p}{x^q}}$.

Answer 1

Simplifying the given expression using the laws of the radical exponent.

$\begin{array}{l}\sqrt[qr]{\frac{x^q}{x^r}}\times \sqrt[rp]{\frac{x^r}{x^p}}\times \sqrt[pq]{\frac{x^p}{x^q}}\\ =\sqrt[qr]{x^{q-r}}\times \sqrt[rp]{x^{r-p}}\times \sqrt[pq]{x^{p-q}}\left[\because \frac{p^m}{p^n}=p^{m-n}\right]\\ ={\left(x^{q-r}\right)}^{\frac{1}{qr}}\times {\left(x^{r-p}\right)}^{\frac{1}{rp}}\times {\left(x^{p-q}\right)}^{\frac{1}{pq}}\left[\because \sqrt[n]{a}=a^{\frac{1}{n}}\right]\\ ={\left(x\right)}^{\frac{q-r}{qr}}\times {\left(x\right)}^{\frac{r-p}{rp}}\times {\left(x\right)}^{\frac{p-q}{pq}}\left[\because {\left(a^m\right)}^n=a^{mn}\right]\\ ={\left(x\right)}^{\frac{q-r}{qr}+\frac{r-p}{rp}+\frac{p-q}{pq}}\left[\because a^{p/q}\times a^{r/s}=a^{\frac{p}{q}+\frac{r}{s}}\right]\\ ={\left(x\right)}^{\frac{p\left(q-r\right)+q\left(r-p\right)+r\left(p-q\right)}{pqr}}\\ ={\left(x\right)}^{\frac{pq-pr+qr-pq+pr-qr}{pqr}}\\ ={\left(x\right)}^{\frac{0}{pqr}}\\ ={\left(x\right)}^0\\ =1\end{array}$

Hence, the simplified value of the given expression is $1$.