The value of `x^d/(x^d + x^(d+b -a)+x^(d+c-a))+(x^d)/(x^d + x^(d+a-b)+x^(d+c-b))+(x^d)/(x^d+x^(d+a-c)+x^(d+b-c))` is

Options

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

Answer

To solve the expression

\[ \frac{x^d}{x^d + x^{d+b-a} + x^{d+c-a}} + \frac{x^d}{x^d + x^{d+a-b} + x^{d+c-b}} + \frac{x^d}{x^d + x^{d+a-c} + x^{d+b-c}}, \]

we will simplify each term step by step.

### Step 1: Rewrite the expression
We can rewrite the expression as:

\[ E = \frac{x^d}{x^d + x^{d+b-a} + x^{d+c-a}} + \frac{x^d}{x^d + x^{d+a-b} + x^{d+c-b}} + \frac{x^d}{x^d + x^{d+a-c} + x^{d+b-c}}. \]

### Step 2: Factor out \(x^d\)
Now, we can factor \(x^d\) out of each denominator:

\[ E = \frac{x^d}{x^d(1 + x^{b-a} + x^{c-a})} + \frac{x^d}{x^d(1 + x^{a-b} + x^{c-b})} + \frac{x^d}{x^d(1 + x^{a-c} + x^{b-c})}. \]

### Step 3: Simplify each term
This simplifies to:

\[ E = \frac{1}{1 + x^{b-a} + x^{c-a}} + \frac{1}{1 + x^{a-b} + x^{c-b}} + \frac{1}{1 + x^{a-c} + x^{b-c}}. \]

### Step 4: Find a common denominator
To combine these fractions, we need a common denominator. The common denominator will be:

\[ (1 + x^{b-a} + x^{c-a})(1 + x^{a-b} + x^{c-b})(1 + x^{a-c} + x^{b-c}). \]

### Step 5: Combine the fractions
Now we can write:

\[ E = \frac{(1 + x^{a-b} + x^{c-b})(1 + x^{a-c} + x^{b-c}) + (1 + x^{b-a} + x^{c-a})(1 + x^{a-c} + x^{b-c}) + (1 + x^{b-a} + x^{c-a})(1 + x^{a-b} + x^{c-b})}{(1 + x^{b-a} + x^{c-a})(1 + x^{a-b} + x^{c-b})(1 + x^{a-c} + x^{b-c})}. \]

### Step 6: Evaluate the numerator
After expanding and simplifying the numerator, we will find that it equals the denominator. Therefore, we have:

\[ E = \frac{(1 + x^{b-a} + x^{c-a})(1 + x^{a-b} + x^{c-b})(1 + x^{a-c} + x^{b-c})}{(1 + x^{b-a} + x^{c-a})(1 + x^{a-b} + x^{c-b})(1 + x^{a-c} + x^{b-c})} = 1. \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{1}. \]