What is the value of `[ (1)/( 1- x ^( (q-p))) + (1)/(1 - x ^(( p-q)))]` ?

Options

A. 0
B. 1
C. `(x ^(q) - x ^(p)) //(x ^(q) + x ^(p))`
D. `(x ^(q) + x ^(p)) //(x ^(q) - ^(p))`

Answer

To solve the expression \[ \frac{1}{1 - x^{(q-p)}} + \frac{1}{1 - x^{(p-q)}} \], we will follow these steps:

### Step 1: Rewrite the expression
The expression can be rewritten using the property of exponents:
\[ \frac{1}{1 - x^{(q-p)}} + \frac{1}{1 - x^{(p-q)}} \]
Notice that \(x^{(p-q)}\) can be rewritten as \(\frac{x^p}{x^q}\) and \(x^{(q-p)}\) can be rewritten as \(\frac{x^q}{x^p}\).

### Step 2: Substitute the rewritten exponents
Substituting these into the expression gives:
\[ \frac{1}{1 - \frac{x^q}{x^p}} + \frac{1}{1 - \frac{x^p}{x^q}} \]

### Step 3: Simplify each term
Now we simplify each term:
1. The first term becomes:
\[ \frac{1}{\frac{x^p - x^q}{x^p}} = \frac{x^p}{x^p - x^q} \]
2. The second term becomes:
\[ \frac{1}{\frac{x^q - x^p}{x^q}} = \frac{x^q}{x^q - x^p} \]

### Step 4: Combine the fractions
Now we can combine the two fractions:
\[ \frac{x^p}{x^p - x^q} + \frac{x^q}{x^q - x^p} \]
Notice that \(x^q - x^p = -(x^p - x^q)\), so we can rewrite the second term:
\[ \frac{x^p}{x^p - x^q} - \frac{x^q}{x^p - x^q} \]

### Step 5: Combine the numerators
Now, combine the numerators over the common denominator:
\[ \frac{x^p - x^q}{x^p - x^q} \]

### Step 6: Simplify the expression
Since the numerator and denominator are the same, we have:
\[ 1 \]

### Final Answer:
The value of the expression is:
\[ 1 \]