The value of `(x^(q)/x^(r))^(1/(qr)) xx (x^(r)/x^(p))^(1/(rp)) xx (x^(p)/x^(q))^(1/(pq)` is equal to:

Options

A. `x^(1/p + 1/q + 1/r)`
B. 0
C. `x^(pq + qr + rp)`
D. 1

Answer

To solve the expression \((\frac{x^q}{x^r})^{\frac{1}{qr}} \cdot (\frac{x^r}{x^p})^{\frac{1}{rp}} \cdot (\frac{x^p}{x^q})^{\frac{1}{pq}}\), we will simplify it step by step.

### Step 1: Simplify Each Term
We start by simplifying each of the three terms in the expression.

1. **First Term:**
\[ \left(\frac{x^q}{x^r}\right)^{\frac{1}{qr}} = \left(x^{q-r}\right)^{\frac{1}{qr}} = x^{\frac{q-r}{qr}} = x^{\frac{1}{r} - \frac{1}{q}} \]

2. **Second Term:**
\[ \left(\frac{x^r}{x^p}\right)^{\frac{1}{rp}} = \left(x^{r-p}\right)^{\frac{1}{rp}} = x^{\frac{r-p}{rp}} = x^{\frac{1}{p} - \frac{1}{r}} \]

3. **Third Term:**
\[ \left(\frac{x^p}{x^q}\right)^{\frac{1}{pq}} = \left(x^{p-q}\right)^{\frac{1}{pq}} = x^{\frac{p-q}{pq}} = x^{\frac{1}{q} - \frac{1}{p}} \]

### Step 2: Combine the Terms
Now we can combine the simplified terms:
\[ x^{\left(\frac{1}{r} - \frac{1}{q}\right)} \cdot x^{\left(\frac{1}{p} - \frac{1}{r}\right)} \cdot x^{\left(\frac{1}{q} - \frac{1}{p}\right)} \]

Using the property of exponents that states \(x^a \cdot x^b = x^{a+b}\), we can combine the exponents:
\[ x^{\left(\frac{1}{r} - \frac{1}{q} + \frac{1}{p} - \frac{1}{r} + \frac{1}{q} - \frac{1}{p}\right)} \]

### Step 3: Simplify the Exponent
Now, let's simplify the exponent:
\[ \frac{1}{r} - \frac{1}{r} + \frac{1}{p} - \frac{1}{p} + \frac{1}{q} - \frac{1}{q} = 0 \]

### Step 4: Final Result
Thus, we have:
\[ x^0 = 1 \]

So, the value of the original expression is:
\[ \boxed{1} \]