${\left(\frac{x^{a^2+b^2}}{x^{ab}}\right)}^{a+b}\cdot {\left(\frac{x^{b^2+c^2}}{x^{bc}}\right)}^{b+c}\cdot {\left(\frac{x^{c^2+a^2}}{x^{ac}}\right)}^{a+c}=$

$={\left(x^{a^2+b^2-ab}\right)}^{a+b}.{\left(x^{b^2+c^2-bc}\right)}^{b+c}.{\left(x^{c^2+a^2-ac}\right)}^{a+c}$

$=x^{(a+b)(a^2+b^2-ab)}\times x^{(b+c)(b^2+c^2-bc)}\times x^{(a+c)(a^2+c^2-ac)}$

$=x^{a^3-b^3}\times x^{b^3-c^3}\times x^{c^3-a^3}$

$=x^0$

$=1$