Question

\[ (x^{a^2+b^2}/x^{2ab})^{a+b}(x^{b^2+c^2}/x^{2bc})^{b+c}(x^{c^2+a^2}/x^{2ca})^{c+a} \]

Solution

\[ = (x^{(a-b)^2})^{a+b}(x^{(b-c)^2})^{b+c}(x^{(c-a)^2})^{c+a} \] \[ = x^{(a-b)^2(a+b)} \cdot x^{(b-c)^2(b+c)} \cdot x^{(c-a)^2(c+a)} \] \[ = x^{(a^3-a^2b-ab^2+b^3) + (b^3-b^2c-bc^2+c^3) + (c^3-c^2a-ca^2+a^3)} \] \[ = x^{2(a^3+b^3+c^3) – (a^2b+ab^2+b^2c+bc^2+c^2a+ca^2)} \] \[ = x^{2(a^3+b^3+c^3)} \]

Answer

\[ \boxed{x^{2(a^3+b^3+c^3)}} \]

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