Simplifying an expression using assumptions [closed]

Question

Is there a way to simplify the following expression

$$A = \frac{\left (a^6-b^6 \right )^2}{c^6-4a^3b^3}$$

assuming

\begin{eqnarray} a = &\ (a-b)^2+b\ (a+1) \cr b = &\ (b-c)^2+c\ (b+1) \cr c = &\ (c-a)^2+a\ (c+1) \end{eqnarray}

Edit

I would like to determine which of these answers for A is correct:

1. $ a^3b^3$
2. $b^3c^3$
3. $a^3b^6$
4. $a^3c^3$
5. $c^6$

Answer 1

Solve[Eliminate[{A == (a^6 - b^6)^2/(c^6 - 4 a^3 b^3), 
    a == (a - b)^2 + b (a + 1), b == (b - c)^2 + c (b + 1), 
    c == (c - a)^2 + a (c + 1)}, {a, b, c}], A] // Simplify

(*  {{A -> -6 - 3*((1/2)*(123 - 
                   55*Sqrt[5]))^(1/3) - 
         3*((1/2)*(123 + 55*Sqrt[5]))^
             (1/3)}, 
   {A -> -6 + (3/2)*(1 + I*Sqrt[3])*
           ((1/2)*(123 - 55*Sqrt[5]))^
             (1/3) + (3 - 3*I*Sqrt[3])/
           (2^(2/3)*(123 - 55*Sqrt[5])^
                (1/3))}, 
   {A -> -6 + (3/2)*(1 - I*Sqrt[3])*
           ((1/2)*(123 - 55*Sqrt[5]))^
             (1/3) + (3 + 3*I*Sqrt[3])/
           (2^(2/3)*(123 - 55*Sqrt[5])^
                (1/3))}}  *)

% // N

(*  {{A -> -21.5225}, {A -> 1.76124 - 12.398 I}, {A -> 
   1.76124 + 12.398 I}}  *)

EDIT: For a multiple choice,

Select[
 {a^3 b^3, b^3 c^3, a^3 b^6, a^3 c^3, c^6},
 Assuming[
   {a == (a - b)^2 + b (a + 1),
    b == (b - c)^2 + c (b + 1),
    c == (c - a)^2 + a (c + 1)},
   Simplify[
    (a^6 - b^6)^2/(c^6 - 4 a^3 b^3) == #]] &]

(*  {c^6}  *)

Answer 2

I'm not certain if this is what you want

Reduce[{A == (a^6 - b^6)^2/(c^6 - 4 a^3 b^3), a == (a - b)^2 + b (a + 1),
  b == (b - c)^2 + c (b + 1), c == (c - a)^2 + a (c + 1)}, A]

tells you that A == -3 (5 + 2 c^2 + c^4) when c== + or - 1 times the square root of any root of $15 + 6 q + 3 q^2 + q^3$. That appears to only have one real root around $-2.78$.