$$\begin{aligned} \text{Expression} &= \left(\frac{x^a}{x^b}\right)^{a+b} \times \left(\frac{x^b}{x^c}\right)^{b+c} \times \left(\frac{x^c}{x^a}\right)^{c+a} \ &= (x^{a-b})^{a+b} \times (x^{b-c})^{b+c} \times (x^{c-a})^{c+a} \quad &\text{(Quotient Rule: } \frac{a^m}{a^n} = a^{m-n}) \ &= x^{(a-b)(a+b)} \times x^{(b-c)(b+c)} \times x^{(c-a)(c+a)} \quad &\text{(Power of a Power Rule)} \ &= x^{a^2-b^2} \times x^{b^2-c^2} \times x^{c^2-a^2} \quad &\text{(Difference of Squares: } (m-n)(m+n) = m^2-n^2) \ &= x^{(a^2-b^2) + (b^2-c^2) + (c^2-a^2)} \quad &\text{(Product Rule)} \ &= x^{a^2-a^2 – b^2+b^2 – c^2+c^2} \ &= x^0 \ &= \mathbf{1} \quad &\text{(Zero Exponent Rule)} \end{aligned}$$

$$\begin{aligned} \text{L.H.S.} &= x^{\frac{a^2}{bc}} \cdot x^{\frac{b^2}{ca}} \cdot x^{\frac{c^2}{ab}} \ &= x^{(\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab})} \quad &\text{(Product Rule)} \ &= x^{\frac{a^2 \cdot a}{bc \cdot a} + \frac{b^2 \cdot b}{ca \cdot b} + \frac{c^2 \cdot c}{ab \cdot c}} \quad &\text{(Find common denominator } abc) \ &= x^{\frac{a^3}{abc} + \frac{b^3}{abc} + \frac{c^3}{abc}} \ &= x^{\frac{a^3+b^3+c^3}{abc}} \ &= x^{\frac{3abc}{abc}} \quad &\text{(Substitute the identity } a^3+b^3+c^3 = 3abc \text{ since } a+b+c=0) \ &= x^3 \ &= \mathbf{x^3} \end{aligned}$$