Evaluate : `((x^(q))/(x^(r )))^((1)/(qr))xx((x^(r ))/(x^(p)))^((1)/(rp))xx((x^(p))/(x^(q)))^((1)/(pq))`

Answer

To evaluate the expression

\[ \left( \frac{x^q}{x^r} \right)^{\frac{1}{qr}} \cdot \left( \frac{x^r}{x^p} \right)^{\frac{1}{rp}} \cdot \left( \frac{x^p}{x^q} \right)^{\frac{1}{pq}}, \]

we will follow these steps:

### Step 1: Apply the Quotient Rule of Exponents
Using the property of exponents that states \(\frac{x^m}{x^n} = x^{m-n}\), we can simplify each fraction inside the parentheses.

\[ \frac{x^q}{x^r} = x^{q - r}, \quad \frac{x^r}{x^p} = x^{r - p}, \quad \frac{x^p}{x^q} = x^{p - q}. \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression using the results from Step 1:

\[ \left( x^{q - r} \right)^{\frac{1}{qr}} \cdot \left( x^{r - p} \right)^{\frac{1}{rp}} \cdot \left( x^{p - q} \right)^{\frac{1}{pq}}. \]

### Step 3: Apply the Power Rule of Exponents
Using the property of exponents that states \((x^m)^n = x^{m \cdot n}\), we can simplify further:

\[ x^{(q - r) \cdot \frac{1}{qr}} \cdot x^{(r - p) \cdot \frac{1}{rp}} \cdot x^{(p - q) \cdot \frac{1}{pq}}. \]

### Step 4: Combine the Exponents
Now, we can combine the exponents since they have the same base \(x\):

\[ x^{\frac{q - r}{qr}} \cdot x^{\frac{r - p}{rp}} \cdot x^{\frac{p - q}{pq}} = x^{\left( \frac{q - r}{qr} + \frac{r - p}{rp} + \frac{p - q}{pq} \right)}. \]

### Step 5: Simplify the Exponent
Now we need to simplify the exponent:

\[ \frac{q - r}{qr} + \frac{r - p}{rp} + \frac{p - q}{pq}. \]

Finding a common denominator, which is \(pqr\):

\[ = \frac{(q - r)p + (r - p)q + (p - q)r}{pqr}. \]

### Step 6: Expand and Combine Terms
Expanding the numerator:

\[ = \frac{qp - rp + rq - pq + pr - qr}{pqr} = \frac{(qp - pq) + (rq - qr) + (pr - rp)}{pqr} = \frac{0}{pqr} = 0. \]

### Step 7: Final Result
Thus, we have:

\[ x^0 = 1. \]

### Final Answer:
The evaluated expression is

\[ \boxed{1}. \]

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