Given:
The equation is: x^(1/p) = y^(1/q) = z^(1/r)
xyz = 1

To Find:
The value of p, q, and r.

Solution:

Step 1: Simplifying the equation
Let's start by simplifying the equation x^(1/p) = y^(1/q) = z^(1/r).

Since all three expressions are equal, we can equate any two of them and solve for the variables.

Let's equate x^(1/p) and y^(1/q):
x^(1/p) = y^(1/q)

Taking the pth power of both sides:
(x^(1/p))^p = (y^(1/q))^p

Simplifying:
x = y^(p/q)

Similarly, let's equate y^(1/q) and z^(1/r):
y^(1/q) = z^(1/r)

Taking the qth power of both sides:
(y^(1/q))^q = (z^(1/r))^q

Simplifying:
y = z^(q/r)

Step 2: Using the given equation xyz = 1
We are given that xyz = 1. Let's substitute the values we derived for x and y in terms of y and z:

x = y^(p/q) = (z^(q/r))^(p/q) = z^(pq/qr) (1)

Multiplying all three equations together:
xyz = x * y * z

Substituting the expressions (1) and (2) in the above equation:
1 = z^(pq/qr) * z^(q/r) * z

Using the properties of exponents:
1 = z^((pq/qr) + (q/r) + 1)

Step 3: Solving for p, q, and r
Since the above equation holds true for any value of z, the exponents must be equal to zero:

(pq/qr) + (q/r) + 1 = 0

Multiplying through by qr:
pq + q^2 + qr = 0

Factoring out q:
q(p + q + r) = 0

Since q cannot be zero (as it appears in the denominator of the original equation), we have two possibilities:

1) p + q + r = 0
2) q = 0

If q = 0, then substituting this value in equation (1), we get:
x = z^(pq/qr) = z^(p*0/r*0) = z^(0/0) = 1

But we are given that xyz = 1, so this implies that x, y, and z are all equal to 1.

Therefore, q cannot be zero.

Hence, the only possible solution is:
p + q + r = 0

This equation does not provide a unique solution for p, q, and r. There are infinitely many combinations of p, q, and r that satisfy this equation.