Question

Prove that:
$[\frac{x^q}{x^r}{]}^{\frac{1}{qr}}\times [\frac{x^r}{x^p}{]}^{\frac{1}{rp}}\times [\frac{x^p}{x^q}{]}^{\frac{1}{pq}}=1$

Answer 1

${\left[\frac{x^q}{x^r}\right]}^{\frac{1}{qr}}\times {\left[\frac{x^r}{x^p}\right]}^{\frac{1}{rp}}\times {\left[\frac{x^p}{x^q}\right]}^{\frac{1}{pq}}$

$={\left(x^{q-r}\right)}^{\frac{1}{qr}}\times {\left(x^{r-p}\right)}^{\frac{1}{rp}}\times {\left(x^{p-q}\right)}^{\frac{1}{pq}}$

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

$=x^{(\frac{q-r}{qr}+\frac{r-p}{rp}+\frac{p-q}{pq})}$

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

$=x^0$

$=1$