Assuming that `x` is a positive real number and `a , b , c` are rational numbers, show that: `((x^b)/(x^c))^a\ ((x^c)/(x^a))^b\ ((x^a)/(x^b))^c=1`

Explanation

Given that
`((x^b)/(x^c))^a\ ((x^c)/(x^a))^b\ ((x^a)/(x^b))^c=1`
Solution
Using laws of exponents
`(a^m)/(a^n)=a^(m-n)`
`a^m xx a^n = a^(m+n)`
`(a^m)n=a^(m xx n)`
`a^-n = 1/a^n`
`((x^b)/(x^c))^a\ ((x^c)/(x^a))^b\ ((x^a)/(x^b))^c`
`(x^(b-c))^a xx (x^(c-b))^b xx (x^(a-b))^c`
`(x^a)^(b-c) xx (x^c)^(c-a) xx (x^c)^(a-b)`
`x^(ab-ac+bc-ab+ca-bc`
`=x^0`
`= 1`
Hence Proved.