To solve the given expression x^3 y^3 z^3 - 3xyz, we need to first simplify the expression x^3 y^3 z^3 and then subtract 3xyz from it.

Given:
x y z = 6 ...(1)
x^2 y^2 z^2 = 18 ...(2)

We can rewrite equation (2) as:
(x y z)^2 = 18
Taking the square root on both sides, we get:
x y z = √18
x y z = 3√2 ...(3)

Simplifying x^3 y^3 z^3:
(x^3 y^3 z^3) = (x y z)^3
(x^3 y^3 z^3) = (3√2)^3 (using equation 3)
(x^3 y^3 z^3) = 27√2 ...(4)

Substituting the values of x y z and x^3 y^3 z^3 in the expression x^3 y^3 z^3 - 3xyz:
(27√2) - 3(6)
27√2 - 18
= 9√2

Hence, the value of x^3 y^3 z^3 - 3xyz is 9√2.

Explanation:
- We are given two equations: x y z = 6 and x^2 y^2 z^2 = 18.
- We simplify equation (2) by taking the square root on both sides, which gives us x y z = 3√2.
- To find x^3 y^3 z^3, we cube x y z, which gives us (x^3 y^3 z^3) = (3√2)^3 = 27√2.
- Finally, we substitute the values of x^3 y^3 z^3 and 3xyz in the given expression and simplify to get the final answer, 9√2.

Note: The formatting and visual appeal of the answer can be enhanced using HTML tags, such as for headings and key points.