Question

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

Answer 1

${\left(\frac{x^a}{x^b}\right)}^{(a+b)}.{\left(\frac{x^b}{x^c}\right)}^{(b+c)}.{\left(\frac{x^c}{x^a}\right)}^{(c+a)}$
$={\left(x^{(a-b)}\right)}^{(a+b)}.{\left(x^{(b-c)}\right)}^{(b+c)}.{\left(x^{(c-a)}\right)}^{(c+a)}$
$=x^{a^2-b^2}.x^{b^2-c^2}.x^{c^2-a^2}$
$=x^{a^2-b^2+b^2-c^2+c^2-a^2}$
$=x^0$
$=1$