Question

\[ (x^{a+b}/x^c)^{a-b}(x^{b+c}/x^a)^{b-c}(x^{c+a}/x^b)^{c-a} \]

Solution

\[ = (x^{a+b-c})^{a-b}(x^{b+c-a})^{b-c}(x^{c+a-b})^{c-a} \] \[ = x^{(a+b-c)(a-b)} \cdot x^{(b+c-a)(b-c)} \cdot x^{(c+a-b)(c-a)} \] \[ = x^{[(a+b-c)(a-b) + (b+c-a)(b-c) + (c+a-b)(c-a)]} \] \[ = x^{(a^2-b^2 – ac + bc + b^2-c^2 – ab + ca + c^2-a^2 – bc + ab)} \] \[ = x^0 \] \[ = 1 \]

Answer

\[ \boxed{1} \]

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