Question

${\left(\frac{x^a}{x^b}\right)}^{a+b-c}\times {\left(\frac{x^b}{x^c}\right)}^{b+c-a}\times {\left(\frac{x^c}{x^a}\right)}^{c+a-b}=$

Answer 1

${\left(\frac{x^a}{x^b}\right)}^{a+b-c}{\left(\frac{x^b}{x^c}\right)}^{b+c-a}{\left(\frac{x^c}{x^a}\right)}^{c+a-b}$

$\Rightarrow x^{(a-b)\cdot (a+b-c)}{x}^{(b-c)\cdot (b+c-a)}{x}^{(c-a)\cdot (c+a-b)}$

$\Rightarrow x^{a^2+ab-ac-ba-b^2+bc}{x}^{b^2+bc-ab-c^2+ac}{x}^{c^2+ac-bc-ac-a^2+ab}$

Since base are same,

$\Rightarrow x^{a^2+ab-ac-ab-b^2+bc+b^2+bc-ab-bc-c^2+ac+c^2+ac-bc-ac-a^2+ab}$

$\Rightarrow x^0$

$=1$