To find the value of a^4 b^4 c^4, we can use the given information and algebraic manipulations.

Given:
abc = 2 ...(1)
a^2 b^2 c^2 = 6 ...(2)
a^3 b^3 c^3 = 8 ...(3)

Let's solve this step by step:

1. Expressing a^4 b^4 c^4 in terms of given values:
We know that (a^4 b^4 c^4) = (a^2 b^2 c^2)^2.
Substituting the value of (a^2 b^2 c^2) from equation (2), we have:
(a^4 b^4 c^4) = (6)^2 = 36.

So, a^4 b^4 c^4 = 36.

2. Finding the value of a^4, b^4, and c^4:
We can rewrite equation (3) as (abc)^3 = 8.
Substituting the value of abc from equation (1), we have:
(2)^3 = 8.

This implies that a^3 b^3 c^3 = 8.

Taking the cube root on both sides, we get:
abc = 2.

Now, let's raise both sides to the power of 2 to get a^2 b^2 c^2:
(a^2 b^2 c^2) = (abc)^2 = (2)^2 = 4.

Similarly, we can raise both sides to the power of 3 to get a^3 b^3 c^3:
(a^3 b^3 c^3) = (abc)^3 = (2)^3 = 8.

3. Solving for individual values of a, b, and c:
From equation (1), we have abc = 2.
From equation (2), we have a^2 b^2 c^2 = 4.
From equation (3), we have a^3 b^3 c^3 = 8.

Taking the square root of equation (2), we get:
√(a^2 b^2 c^2) = √4,
which simplifies to a b c = 2.

Taking the cube root of equation (3), we get:
∛(a^3 b^3 c^3) = ∛8,
which simplifies to a b c = 2.

Hence, a b c = 2.

4. Using the value of a b c to find a^4 b^4 c^4:
We already established that a b c = 2.
Therefore, substituting this value into the expression for a^4 b^4 c^4, we have:
(a^4 b^4 c^4) = (a^2 b^2 c^2)^2 = (4)^2 = 16.

So, a^4 b^4 c^4 = 16.

Therefore, the value of a^4 b^4 c^4 is 16.