**Problem Analysis:**
We are given two equations:
1. x * y = 12
2. x * y = 27

We need to find the value of x³ * y³.

**Solution:**

Let's solve the given equations step by step to find the value of x³ * y³.

1. Solving equation 1:
x * y = 12

We can rewrite this equation as:
xy = 12

Now, let's solve this equation to find the values of x and y.

By dividing both sides of the equation by y, we get:
x = 12/y

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

Simplifying, we get:
12 = 12

This equation is always true, which means there are infinite solutions for x and y. Therefore, the given equations are inconsistent.

2. Solving equation 2:
x * y = 27

Similarly, we can rewrite this equation as:
xy = 27

Let's solve this equation to find the values of x and y.

By dividing both sides of the equation by y, we get:
x = 27/y

Substituting this value of x in the equation xy = 27, we get:
(27/y) * y = 27

Simplifying, we get:
27 = 27

This equation is also always true, which means there are infinite solutions for x and y. Therefore, the given equations are inconsistent.

**Conclusion:**
Based on the given equations, we can see that there are no specific values of x and y that satisfy both equations simultaneously. Therefore, we cannot determine the value of x³ * y³.