Question

Prove that : `((x^a)/(x^b))^( a + b - c ) (( x^b)/(x^c))^( b + c - a )((x^c)/(x^a))^( c + a - b)`

Answer 1

LHS = `((x^a)/(x^b))^( a + b - c ) (( x^b)/(x^c))^( b + c - a )((x^c)/(x^a))^( c + a - b)`

= `(x^(a - b))^( a + b - c) xx (x^(b - c))^( b + c - a ) xx ( x^(c - a ))^( c + a - b)`

= `x^[( a - b )( a + b - c )] xx x^[( b - c )( b + c - a )] xx x^[( c - a )(c + a - b)]`

= `x^( a^2 + ab - ac - ab - b^2 + bc) xx x^(b^2 + bc - ab - cd - c^2 + ac) xx x^( c^2 + ac - bc - ac - a^2 + ab )`

= `x^(a^2 - ac - b^2 + bc + b^2 - ab - c^2 + ac + c^2 - bc - a^2 + ab )`
= x⁰
= 1
= RHS