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Question

Evaluate \((x^{b-c})^{b+c-a} (x^{c-a})^{c+a-b} (x^{a-b})^{a+b-c}\)

Solution:

\(x^{(b-c)(b+c-a)} \cdot x^{(c-a)(c+a-b)} \cdot x^{(a-b)(a+b-c)}\)

\( = x^{(b-c)(b+c) + (b-c)(-a)} \cdot x^{(c-a)(c+a) + (c+a)(-b)} \cdot x^{(a-b)(a+b) + (a+b)(-c)}\)

\(= x^{(b^2 – c^2 – ba + ca)} \cdot x^{(c^2 – a^2 – cb + ab)} \cdot x^{(a^2 – b^2 – ac + bc)}\)

\( = x^{(b^2 – c^2 – ba + ca) + (c^2 – a^2 – cb + ab) + (a^2 – b^2 – ac + bc)}\)

\( = x^0 = 1\)

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