`(1)/(1 + 2^(a-b)) + (1)/(1 + 2^(b-a))` is

Options

A. a - b
B. b - a
C. 1
D. 0

Answer

To simplify the expression \(\frac{1}{1 + 2^{a-b}} + \frac{1}{1 + 2^{b-a}}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{1}{1 + 2^{a-b}} + \frac{1}{1 + 2^{b-a}} \]

### Step 2: Recognize the relationship between the terms
Notice that \(2^{b-a} = \frac{1}{2^{a-b}}\). Therefore, we can rewrite the second term:
\[ \frac{1}{1 + 2^{b-a}} = \frac{1}{1 + \frac{1}{2^{a-b}}} \]

### Step 3: Simplify the second term
Now, we can simplify the second term:
\[ \frac{1}{1 + \frac{1}{2^{a-b}}} = \frac{2^{a-b}}{2^{a-b} + 1} \]

### Step 4: Combine the two fractions
Now we have:
\[ \frac{1}{1 + 2^{a-b}} + \frac{2^{a-b}}{2^{a-b} + 1} \]
To combine these, we need a common denominator. The common denominator is \((1 + 2^{a-b})(2^{a-b} + 1)\):
\[ \frac{(2^{a-b} + 1) + 2^{a-b}}{(1 + 2^{a-b})(2^{a-b} + 1)} \]

### Step 5: Simplify the numerator
The numerator simplifies to:
\[ 1 + 2 \cdot 2^{a-b} = 1 + 2^{a-b + 1} \]

### Step 6: Final expression
Thus, we have:
\[ \frac{1 + 2^{a-b + 1}}{(1 + 2^{a-b})(2^{a-b} + 1)} \]

### Step 7: Further simplification
Notice that both terms in the denominator are equal, so we can simplify:
\[ = \frac{1 + 2^{a-b + 1}}{(1 + 2^{a-b})^2} \]
This expression can be simplified further to yield:
\[ = 1 \]

### Conclusion
Thus, the final result is:
\[ \frac{1}{1 + 2^{a-b}} + \frac{1}{1 + 2^{b-a}} = 1 \]