Question

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))`?

Answer 1

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))`