Find the value of `(1)/(1+x^(a-b))+(1)/(1+x^(b-a))`

Options

A. 0
B. `-1`
C. 1
D. `x^(a+b)`

Answer

To solve the problem \( \frac{1}{1+x^{a-b}} + \frac{1}{1+x^{b-a}} \), we can follow these steps:

### Step 1: Rewrite the Terms
We start with the expression:
\[ \frac{1}{1+x^{a-b}} + \frac{1}{1+x^{b-a}} \]

### Step 2: Recognize the Relationship Between Exponents
Notice that \( x^{b-a} = \frac{1}{x^{a-b}} \). This means we can rewrite the second term:
\[ \frac{1}{1+x^{b-a}} = \frac{1}{1+\frac{1}{x^{a-b}}} \]

### Step 3: Simplify the Second Term
Now simplify the second term:
\[ \frac{1}{1+\frac{1}{x^{a-b}}} = \frac{x^{a-b}}{x^{a-b}+1} \]

### Step 4: Rewrite the Original Expression
Now we can rewrite the original expression using this simplification:
\[ \frac{1}{1+x^{a-b}} + \frac{x^{a-b}}{x^{a-b}+1} \]

### Step 5: Find a Common Denominator
The common denominator for the two fractions is \( (1+x^{a-b})(x^{a-b}+1) \):
\[ \frac{(x^{a-b}+1) + (1+x^{a-b})}{(1+x^{a-b})(x^{a-b}+1)} \]

### Step 6: Combine the Numerator
Combine the terms in the numerator:
\[ \frac{(x^{a-b}+1) + (1+x^{a-b})}{(1+x^{a-b})(x^{a-b}+1)} = \frac{2(x^{a-b}+1)}{(1+x^{a-b})(x^{a-b}+1)} \]

### Step 7: Simplify the Expression
Notice that \( 1+x^{a-b} \) and \( x^{a-b}+1 \) are the same, so:
\[ \frac{2(x^{a-b}+1)}{(1+x^{a-b})^2} \]

### Step 8: Final Result
Thus, we can conclude that:
\[ \frac{1}{1+x^{a-b}} + \frac{1}{1+x^{b-a}} = 1 \]

### Final Answer
The value of the expression is \( 1 \).

---