Prove: \[ \left(\frac{x^a}{x^b}\right)^{a^2+ab+b^2} \times \left(\frac{x^b}{x^c}\right)^{b^2+bc+c^2} \times \left(\frac{x^c}{x^a}\right)^{c^2+ca+a^2} = 1 \]
Proof
\[ = x^{(a-b)(a^2+ab+b^2)} \cdot x^{(b-c)(b^2+bc+c^2)} \cdot x^{(c-a)(c^2+ca+a^2)} \]
\[ = x^{(a^3-b^3) + (b^3-c^3) + (c^3-a^3)} \]
\[ = x^{0} \]
\[ = 1 \]