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1. ${(\frac{x^{b}}{x^{c}})}^{(b + c - a)} .$ ${(\frac{x^{c}}{x^{a}})}^{(c + a - b)} .$ ${(\frac{x^{a}}{x^{b}})}^{(a + b - c)}$ = ?

A. x^abc

B. 1

C. x^{ab + bc + ca}

D. x^{a + b + c}

Solution:

\begin{aligned} Given Exp . \\ = x^{(b - c) (b + c - a)} . x^{(c - a) (c + a - b)} . x^{(a - b) (a + b - c)} \end{aligned}

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

x^{(c - a) (c + a) - b (c - a)} .

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

\begin{aligned} = x^{(b^{2} - c^{2} + c^{2} - a^{2} + a^{2} - b^{2})} . x^{- a (b - c) - b (c - a) - c (a - b)} \\ = x^{0} . x^{(- a b + a c - b c + b a - c a + c b)} \\ = (x^{0} \times x^{0}) \\ = (1 \times 1) \\ = 1 \end{aligned}

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