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The value of $$ \left(\frac{x^a}{x^b}\right)^{(a^2 + ab + b^2)} \left(\frac{x^b}{x^c}\right)^{(b^2 + bc + c^2)} \left(\frac{x^c}{x^a}\right)^{(c^2 + ca + a^2)}$$ is:
Given, $$ \left(\frac{x^a}{x^b}\right)^{(a^2 + ab + b^2)} \left(\frac{x^b}{x^c}\right)^{(b^2 + bc + c^2)} \left(\frac{x^c}{x^a}\right)^{(c^2 + ca + a^2)}$$
= $$x^{\left(a-b\right)\left(a^2+ab+b^2\right)}\times\ x^{\left(b-c\right)\left(b^2+bc+c^2\right)}\times\ x^{\left(c-a\right)\left(c^2+ac+a^2\right)}$$
= $$x^{a^3-b^3}\times\ x^{b^3-c^3}\times\ x^{c^3-a^3}=x^{a^3-b^3+b^3-c^3+c^3-a^3}=x^0=1$$
So, the correct answer is option C.
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