Given Information: ab bc ca=0
To Find: (1/a²-bc ) (1/b²-ca) (1/c²-ac)

Solution:

Step 1: Factorize ab bc ca=0.
Explanation: Since the product of two or more factors is equal to zero only if one or more of the factors is equal to zero, therefore we can write ab bc ca=0 as:

ab = 0 or bc = 0 or ca = 0

Step 2: Solve for a, b, c in each equation.
Explanation:

If ab = 0, then either a = 0 or b = 0.
If bc = 0, then either b = 0 or c = 0.
If ca = 0, then either c = 0 or a = 0.

Step 3: Substitute the values of a, b, c in the expression (1/a²-bc ) (1/b²-ca) (1/c²-ac) and simplify.
Explanation:

If a = 0, then (1/a²-bc ) = undefined.
If b = 0, then (1/b²-ca) = undefined.
If c = 0, then (1/c²-ac) = undefined.

Therefore, the expression (1/a²-bc ) (1/b²-ca) (1/c²-ac) is undefined if any of the variables a, b, or c is equal to zero.

If none of the variables is equal to zero, then we can substitute the values of a, b, and c in the expression and simplify:

(1/a²-bc ) (1/b²-ca) (1/c²-ac) = (1/a²) (1/b²) (1/c²)

= 1/(a² b² c²)

Step 4: Write the final answer.
Explanation: The value of (1/a²-bc ) (1/b²-ca) (1/c²-ac) is 1/(a² b² c²) if none of the variables a, b, or c is equal to zero. Otherwise, the expression is undefined.

Therefore, the final answer is:

(1/a²-bc ) (1/b²-ca) (1/c²-ac) = 1/(a² b² c²) (if none of the variables a, b, or c is equal to zero)

Conclusion: The value of (1/a²-bc ) (1/b²-ca) (1/c²-ac) is 1/(a² b² c²) if none of the variables a, b, or c is equal to zero. Otherwise, the expression is undefined.