Question

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

Answer 1

$\left(\frac{x^a}{x^b}{)}^{a+b-c}\right(\frac{x^b}{x^c}{)}^{b+c-a}(\frac{x^c}{x^a}{)}^{c+a-b}$
$={x^{a-b}}^{a+b-c}{x^{b-c}}^{b+c-a}{x^{c-a}}^{c+a-b}$
$=x^{a^2-b^2-ac+bc}{x}^{b^2-c^2-ab+ca}{x}^{c^2-a^2-bc+ba}$
By using law $a^m\times a^n=a^{m+n}$, we get $x^{a^2-b^2-ac+bc+b^2-c^2-ab+ca+c^2-a^2-bc+ba}$
$=x^0$
$=1$