If m is the geometric mean of ( y z )^ ( yz ) , ( z x )^ ( zx ) and ( x y )^ ( xy ) then what is the value of m ?

Step-by-step solution

Three terms are G₁= ( y z )^ y z G₂= ( z x )^ z x G₃= ( x y )^ x y Geometric mean of three terms is aligned & m= [3] G₁ G₂ G₃ ...(1) & aligned G₁ G₂ G₃ & = ( y z )^ y z ( z x )^ z x ( x y )^ x y & = y^ y y^ z z^ y z^ z z^ z z^ x x^ z x^ x x^ x x^ y y^ x y^ y & = ( y x )^ z ( z y )^ x ( x z )^ y aligned aligned Taking log both sides aligned G₁ G₂ G₃= & [ ( y x )^ z ]+ [ ( z y )^ x ]+ [ ( x z )^ y ] = & z y- z x+ x z - x y+ y x- y z aligned aligned & G₁ G₂ G₃=0 & G₁ G₂ G₃= e ^0=1 & Hence m= [3] G₁ G₂ G₃ =(1)^ 1 3 & m