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

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Solution

As given $(x^{a + b})^{a - b} . (x^{b + c})^{b - c} . (x^{c + a})^{c - a}$
$\Rightarrow (x^{(a + b) \cdot (a - b)}) . (x^{(b + c) \cdot (b - c)}) . (x^{(c + a) \cdot (c - a)})$
$\Rightarrow (x^{a^{2} + a b - a b - b^{2}}) . (x^{b^{2} + b c - b c - c^{2}}) . (x^{c^{2} + a c - a c - a^{2}})$
$\Rightarrow (x^{a^{2} - b^{2}}) . (x^{b^{2} - c^{2}}) . (x^{c^{2} - a^{2}})$
$\Rightarrow (x^{a^{2} - b^{2} + b^{2} - c^{2} + c^{2} - a^{2}})$
$\Rightarrow x^{0}$
$∵ a^{0} = 1$
$∴ x^{0} = 1$

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