Solution:

To find the values of xy, yz, and zx, we need to solve the given equations and find the values of x, y, and z.

Given:
xy = 12 ...(1)
x * 2 y * 2 z * 2 = 64 ...(2)

Let's solve these equations step by step:

Solving Equation (1):

From equation (1), we have xy = 12.

To find the values of x and y, we can rewrite the equation as:
x = 12/y ...(3)

Substituting this value of x in equation (1), we get:
(12/y) * y = 12
12 = 12

This equation holds true, which means any value of y can satisfy it. Therefore, there are infinitely many possible values of x and y that satisfy equation (1).

Solving Equation (2):

From equation (2), we have x * 2 y * 2 z * 2 = 64.

Expanding the equation, we get:
2 * 2 * 2 * x * y * z = 64
8xyz = 64

Dividing both sides of the equation by 8, we have:
xyz = 8 ...(4)

So, the value of xyz is 8.

Finding xy:

From equation (1), we know that xy = 12. Therefore, xy = 12.

Finding yz:

Substituting the value of xyz from equation (4) into the equation yz, we have:
yz = 8/y ...(5)

Since the value of y is not specified, we cannot determine the exact value of yz. It can take different values depending on the value of y.

Finding zx:

Similarly, substituting the value of xyz from equation (4) into the equation zx, we have:
zx = 8/x ...(6)

Since the value of x is not specified, we cannot determine the exact value of zx. It can take different values depending on the value of x.

To summarize:
- xy = 12
- yz = 8/y (where y can take any value)
- zx = 8/x (where x can take any value)

Therefore, the values of xy, yz, and zx are not uniquely determined by the given equations. They can have different values depending on the values of x and y.