Question

Simplify:

`(x^a/x^b)^(a + b) (x^b/x^c)^(b + c) (x^c/x^a)^(c + a)`

Answer 1

The given expression is `(x^a/x^b)^(a + b) (x^b/x^c)^(b + c) (x^c/x^a)^(c + a)`

= `(x^(a - b))^(a + b) (x^(b - c))^(b + c) (x^(c - a))^(c + a)`  ...`(∵ a^m/a^n = a^(m - n))`

= `x^((a + b)(a - b)) x^((b - c)(b + c)) x^((c - a)(c + a))`  ...(∵ (a^m)ⁿ = a^mn)

= `x^(a^2 - b^2) x^(b^2 - c^2)x^(c^2 - a^2)`

= `x^(a^2 - b^2 + b^2 - c^2 + c^2 - a^2)`  ...(∵ a^m × aⁿ = a^{m + n})

= x⁰ 

= 1

Hence the required is 1.