If# (y/z)^a * (z/x)^b*(x/y)^c = 1# Then prove #(y/z)^(1/(b-c))=(z/x)^(1/(c-a)) = (x/y)^(1/(a-b))#?

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Answer 1 · 2018-05-08T11:20:51

Please see below.

As #(y/z)^a*(z/x)^b*(x/y)^c=1#

#x^(c-b)y^(a-c)z^(b-a)=1# .........(A)

Let #(y/z)^(1/(b-c))=k#

then #y=k^(b-c)z# and putting this in (A), we get

#x^(c-b)k^((b-c)(a-c))z^(a-c)z^(b-a)=1#

or #x^(c-b)k^((b-c)(a-c))z^(b-c)=1#

or #(z/x)^(b-c)=k^((b-c)(c-a))# (-note change from #a-c# to #c-a#)

or #z/x=k^(c-a)#

or #(z/x)^(1/(c-a))=k#

Similarly #(x/y)^(1/(a-b))=k#

Hence #(y/z)^(1/(b-c))=(z/x)^(1/(c-a))=(x/y)^(1/(a-b))#