Question

Prove the following:
${\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}\times {\left(\frac{x^b}{x_c}\right)}^{b^2+bc+c^2}\times {\left(\frac{x^c}{x^a}\right)}^{c^2+ca+a^2}=1$

Answer 1

${\left(\frac{x^a}{x^b}\right)}^{a^2+ab+b^2}\times {\left(\frac{x^b}{x_c}\right)}^{b^2+bc+c^2}\times {\left(\frac{x^c}{x^a}\right)}^{c^2+a^2+ca}=x^{(a-b)(a^2+b^2+ab)}\times x^{(b-c)(b^2+bc+c^2)}\times x^{(c-a)(c^2+a^2+ca)}=x^{(a^3-b^3)}\times x^{(b^3-c^3)}\times x^{(c^3-a^3)}=x^{a^3-b^3+b^3-c^3+c^3-a^3}=x^{\circ }=1$