ABC26GN1883 · Surds and Indices

Given Expression $=\left\{x^{(a-b)}\right\}^{\left(a^{2}+b^{2}+a b\right)} \cdot\left\{x^{(b-c)}\right\}^{\left(b^{2}+c^{2}+b c\right)} \cdot\left\{x^{(c-a)}\right\}^{\left(c^{2}+a^{2}+c a\right)}$ $$\begin{aligned} & =x^{(a-b)\left(a^{2}+b^{2}+a b\right)} \cdot x^{(b-c)\left(b^{2}+c^{2}+b c\right)} \cdot x^{(c-a)\left(c^{2}+a^{2}+c a\right)} \\ & =x^{\left(a^{3}-b^{3}\right)} \cdot x^{\left(b^{3}-c^{3}\right)} \cdot x^{\left(c^{3}-a^{3}\right)}=x^{\left(a^{3}-b^{3}+b^{3}-c^{3}+c^{3}-a^{3}\right)}=x^{0}=1 . \end{aligned}$$