Given:
(x^(1/a-b))^(1/a-c) * (x^(1/b-c))^(1/b-a) * (x^(1/c-a))^(1/c-a)

To prove:
The given expression is equal to 1.

Proof:
Let's simplify the given expression step by step.

Step 1:
We can simplify each term inside the parentheses using the exponent rule:
(x^(1/a-b))^(1/a-c) = x^((1/a-b) * (1/a-c))
(x^(1/b-c))^(1/b-a) = x^((1/b-c) * (1/b-a))
(x^(1/c-a))^(1/c-b) = x^((1/c-a) * (1/c-b))

Step 2:
Now, let's simplify the exponent expressions:
(1/a-b) * (1/a-c) = (1/a - b - c + bc) / (a - c)
(1/b-c) * (1/b-a) = (1/b - c - a + ac) / (b - a)
(1/c-a) * (1/c-b) = (1/c - a - b + ab) / (c - b)

Step 3:
Substitute the simplified exponent expressions back into the main expression:
x^((1/a - b - c + bc) / (a - c)) * x^((1/b - c - a + ac) / (b - a)) * x^((1/c - a - b + ab) / (c - b))

Step 4:
Combine the terms with the same base (x) by adding the exponents:
x^(((1/a - b - c + bc) / (a - c)) + ((1/b - c - a + ac) / (b - a)) + ((1/c - a - b + ab) / (c - b)))

Step 5:
Now, let's simplify the expression inside the exponent:
(((1/a - b - c + bc) / (a - c)) + ((1/b - c - a + ac) / (b - a)) + ((1/c - a - b + ab) / (c - b)))
= ((b - c + bc - ab) / (a - c)) + ((c - a + ac - bc) / (b - a)) + ((a - b + ab - ac) / (c - b))
= (a - b + ab - ac) / (c - b) + (b - c + bc - ab) / (a - c) + (c - a + ac - bc) / (b - a)
= (a - b + ab - ac + b - c + bc - ab + c - a + ac - bc) / (c - b)(a - c)(b - a)
= (0) / (c - b)(a - c)(b - a)
= 0

Conclusion:
Therefore, the given expression is equal to 1.