Solution:

To find the value of 1/x * 1/y * 1/z, we need to simplify the given equation and express everything in terms of x.

Let's start by taking the logarithm of both sides of the equation:

log(2^x) = log(3^y) = log(12^(-z))

Using the logarithmic property log(a^b) = b*log(a), we can rewrite the equation as:

x*log(2) = y*log(3) = -z*log(12)

Now, let's solve for x:

x = (y*log(3)) / log(2)

Similarly, solving for y:

y = (x*log(2)) / log(3)

And solving for z:

z = -(x*log(2)) / log(12)

We now have expressions for x, y, and z in terms of each other.

To find the value of 1/x * 1/y * 1/z, we can substitute these expressions back into the equation:

1/x * 1/y * 1/z = 1 / ((y*log(3)) / log(2)) * 1 / ((x*log(2)) / log(3)) * 1 / (-(x*log(2)) / log(12))

Simplifying the expression:

1/x * 1/y * 1/z = (log(2) * log(3) * log(12)) / (y * x * log(2) * log(3) * log(12))

The log(2), log(3), and log(12) terms cancel out, leaving us with:

1/x * 1/y * 1/z = 1 / (y * x)

Therefore, the value of 1/x * 1/y * 1/z is 1 / (y * x).

In summary, the value of 1/x * 1/y * 1/z is 1 / (y * x).