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))?

1 Answer
May 8, 2018

Please see below.

Explanation:

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))