Question

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Answer 1

Given :

$\sf \bullet \: \: { (\dfrac{ \: {x}^{b} }{{x}^{c} } )}^{(b + c + a)} \times { (\dfrac{ \: {x}^{c} }{{x}^{a} } )}^{( c + a + b)} \times { (\dfrac{ \: {x}^{a} }{{x}^{b} } )}^{(a + b + c )}$

Identity used :

$\sf \bullet \: \dfrac{ {p}^{m} }{ {q}^{n} } \: = \: {p}^{m} - {q}^{n} \\ \\ \sf \bullet \: \: {p}^{m} \times {p}^{n} = \: {p}^{(m + n)} \\ \\ \sf \bullet \: \: {( {p}^{m}) }^{n} \: = \: {p}^{m \times n} \\ \\ \sf \bullet \: {p}^{0} = 1$

Solution :

$\sf \implies \: \: { ( {x}^{(b - c)} )}^{(b + c + a)} \times { ( {x}^{(c - a)} )}^{( c + a + b)} \times { ( {x}^{(a - b)} )}^{(a + b + c )} \\ \\ \sf \implies \: \: {x}^{(b - c)(b + c + a)} \: \: \times \: \: {x}^{(c - a)( c + a + b)} \: \: \times \: \: {x}^{(a - b)(a + b + c)} \\ \\ \sf \implies \: {x}^{( \: \: (b - c)(b + a + c) \: + \:( c - a)(c + a + b) \: + \: (a - b)(a + b + c) \: \: \: ) } \\ \\ \sf \bullet \: rearranging \: (a + b + c) \\ \\ \sf \implies \: {x}^{( \: \: (b - c)(a + b + c) \: + \:( c - a)( a + b + c) \: + \: (a - b)(a + b + c) \: \: \: )} \\ \\ \sf \bullet \: taking \: (a + b + c) \: common \: \\ \\ \sf \implies \: {x}^{( \: \: (a + b + c) \times (b - c \: + \: c - a \: + \: a - b) \: \: )} \\ \\ \sf \implies \: {x}^{(a + b + c) \times \: 0} \\ \\ \sf \implies \: {x}^{ \: 0} \\ \\ \sf \implies \bold{1}$

ANSWER : 1