Introduction:
In this problem, we are given three equations: a^x = b, b^y = c, and c^z = a. We need to find the value of xyz.

Solution:

Step 1: Find the relationship between exponents

From the given equations, we can observe that x, y, and z are exponents. To find the relationship between these exponents, we can substitute the given equations into each other.

Step 2: Substitute equations to find relationship between exponents

Substituting the first equation into the second equation, we get:
(b^y) = c
Replacing b with a^x, we have:
(a^x)^y = c
Using the property of exponents, we can simplify this to:
a^(xy) = c

Similarly, substituting the second equation into the third equation, we get:
(c^z) = a
Replacing c with b^y, we have:
(b^y)^z = a
Using the property of exponents, we can simplify this to:
b^(yz) = a

Step 3: Equate the two expressions for a

Since both expressions are equal to a, we can equate them:
a^(xy) = b^(yz)

Step 4: Equate the exponents

Since the base is the same, the exponents must also be equal:
xy = yz

Step 5: Simplify the equation

Dividing both sides of the equation by y:
x = z

Step 6: Substitute x = z into the equation

Substituting x = z into the equation xy = yz, we get:
xz = yz

Step 7: Cancel out the common terms

Dividing both sides of the equation by z:
x = y

Step 8: Substitute x = y = z into the equation

Substituting x = y = z into the equation xyz, we get:
xyz = xxx
xyz = x^3

Step 9: Find the value of xyz

Since x = y = z, we can substitute any value for x, y, or z. Let's assume x = 1:
xyz = 1^3
xyz = 1

Conclusion:
The value of xyz is 1.