$a = \frac{x^p}{x^q}, b = \frac{x^q}{x^r}$ এবং $c = \frac{x^r}{x^p}$

দেওয়া আছে, $a = \frac{x^p}{x^q}, b = \frac{x^q}{x^r}$ এবং $c = \frac{x^r}{x^p}$

প্রদত্ত রাশি, abc = $\frac{x^p}{x^q}\times \frac{x^q}{x^r}\times \frac{x^r}{x^p}=1$

দেওয়া আছে, $a = \frac{x^p}{x^q}, b = \frac{x^q}{x^r}$ এবং $c = \frac{x^r}{x^p}$

বামপক্ষ = $a^{\frac{1}{{}^{pq}} }\times b^{\frac{1}{qr}}\times c^{\frac{1}{rp}}$

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

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

= $x^{\frac{pr-qr+pq-pr+qr-pq}{pqr}}$

= $x^{\frac{0}{pqr}}$

= x° = 1 = ডানপক্ষ

$a^{\frac{1}{{}^{pq}} }\times b^{\frac{1}{qr}}\times c^{\frac{1}{rp}}$ = 1 (প্রমাণিত)

দেওয়া আছে, $a = \frac{x^p}{x^q}, b = \frac{x^q}{x^r}$ এবং $c = \frac{x^r}{x^p}$

বামপক্ষ = (p+q) log a + (q+r) log b + (r+p) log c

= (p+q) log $\frac{x^p}{x^q}$ + (q+r) log $\frac{x^q}{x^r}$ + (r+p) log $\frac{x^r}{x^p}$

= $(p+q) \log x^{p-q} + (q+r) \log x^{q-r} + (r+p) \log x^{r-p}$

= $\log x^{p^2-q^2} + \log x^{q^2-r^2} + \log x^{r^2-p^2}$

= $\log \left(x{ }^{p^2-q^2+q^2-r^2+r^2-p^2}\right)$

= log x° = 0

= ডানপক্ষ

(p+q) log a + (q+r) log b + (r+p) log = 0 (দেখানো হলো )