**Solution:**

To find the values of a, b, and c that would reduce the given expression to zero, we need to simplify the expression and solve for a, b, and c. Let's break down the problem step by step:

**Step 1: Simplifying the Expression**

We are given the expression:

\[X^{a^2(b^{-1})(c^{-1})}. X^{b^2(c^{-1})(a^{-1})}.X^{c^2(a^{-1})(b^{-1})} - X^3\]

To simplify this expression, we can use the properties of exponents and algebraic manipulations. Let's simplify each term separately:

Term 1: \(X^{a^2(b^{-1})(c^{-1})}\)

- Using the property \(a^{-1} = \frac{1}{a}\), we can rewrite the exponent as \(\frac{1}{b} \cdot \frac{1}{c}\).
- Using the property \((a^m)^n = a^{mn}\), we can simplify the expression to \(X^{a^2/bc}\).

Term 2: \(X^{b^2(c^{-1})(a^{-1})}\)

- Using the property \(a^{-1} = \frac{1}{a}\), we can rewrite the exponent as \(\frac{1}{c} \cdot \frac{1}{a}\).
- Using the property \((a^m)^n = a^{mn}\), we can simplify the expression to \(X^{b^2/ca}\).

Term 3: \(X^{c^2(a^{-1})(b^{-1})}\)

- Using the property \(a^{-1} = \frac{1}{a}\), we can rewrite the exponent as \(\frac{1}{a} \cdot \frac{1}{b}\).
- Using the property \((a^m)^n = a^{mn}\), we can simplify the expression to \(X^{c^2/ab}\).

Now, let's combine these simplified terms:

\[X^{a^2/bc} \cdot X^{b^2/ca} \cdot X^{c^2/ab} - X^3\]

Using the property \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents:

\[X^{(a^2/bc) + (b^2/ca) + (c^2/ab)} - X^3\]

**Step 2: Equating the Expression to Zero**

To reduce the expression to zero, we need to find the values of a, b, and c that make the exponent of X equal to zero:

\[(a^2/bc) + (b^2/ca) + (c^2/ab) = 3\]

To simplify this equation, we can multiply both sides by the common denominator abc:

\[a^3 + b^3 + c^3 = 3abc\]

This equation represents a cubic equation in a, b, and c. To solve this equation, we can use various methods such as factoring, synthetic division, or numerical methods like Newton's method. Unfortunately, without any additional information or constraints, it is not possible to determine the specific values of a, b, and c that would make the expression equal to