Simplify `((p^(a))/(p^(b))) xx ((p^(b))/(p^(c ))) xx ((p^(c ))/(p^(a)))^(b)`

Answer

To simplify the expression \(\frac{p^a}{p^b} \times \frac{p^b}{p^c} \times \left(\frac{p^c}{p^a}\right)^b\), we will use the properties of indices. Let's break it down step by step.

### Step 1: Simplify each fraction using the property of indices
Using the property \(\frac{p^m}{p^n} = p^{m-n}\), we can simplify each term:

1. \(\frac{p^a}{p^b} = p^{a-b}\)
2. \(\frac{p^b}{p^c} = p^{b-c}\)
3. \(\left(\frac{p^c}{p^a}\right)^b = \left(p^{c-a}\right)^b = p^{b(c-a)} = p^{bc-ab}\)

### Step 2: Combine the simplified terms
Now we can combine all the simplified terms:

\[ p^{a-b} \times p^{b-c} \times p^{bc-ab} \]

Using the property \(p^m \times p^n = p^{m+n}\), we can add the exponents:

\[ p^{(a-b) + (b-c) + (bc-ab)} \]

### Step 3: Simplify the exponent
Now, let's simplify the exponent:

\[ (a-b) + (b-c) + (bc-ab) = a - b + b - c + bc - ab \]

The \(b\) and \(-b\) cancel each other out:

\[ = a - c + bc - ab \]

### Step 4: Write the final simplified expression
Thus, we can write the final simplified expression as:

\[ p^{(a - c + bc - ab)} \]

### Summary of the steps:
1. Simplified each fraction using the property of indices.
2. Combined the simplified terms using the multiplication property of indices.
3. Simplified the exponent by combining like terms.