Question

Simplify:

`((x^(a + b))^3 * (x^(b + c))^3 * (x^(c + a))^3)/((x^a x ^b x^c)^6`

Answer 1

We are asked to simplify:

`((x^(a + b))^3 * (x^(b + c))^3 * (x^(c + a))^3)/((x^a * x ^b * x^c)^6`

Step 1: Expand numerator

`(x^(a + b))^3 = x^(3(a + b))`,

`(x^(b + c))^3 = x^(3(b + c))`,

`(x^(c + a))^3 = x^(3(c + a))`

So numerator:

`x^(3(a + b)) * x^(3(b + c)) * x^(3(c + a)) = x^((3(a + b) + 3(b + c) + 3(c + a)`

Simplify exponents:

3a + 3b + 3b + 3c + 3c + 3a = 6a + 6b + 6c

So numerator = `x^(6a + 6b + 6c)`

Step 2: Expand denominator

`(x^a * x^b * x^c)^6 = (x^(a + b + c))^6`

= `x^(6(a + b + c))`

= `x^(6a + 6b + 6c)`

Step 3: Simplify fraction

`x^(6a + 6b + 6c)/(x^(6a + 6b + 6c)) = x^0 = 1`