To evaluate the expression [tex]\(\frac{x^p}{x^p + x^q} + \frac{1}{x^{p - q} + 1}\)[/tex], let’s break it down step-by-step.
Given:
[tex]\[ \frac{x^p}{x^p + x^q} + \frac{1}{x^{p - q} + 1} \][/tex]
Let's consider the two fractions separately, then we will combine them to find a common denominator and simplify.
1. The first fraction is:
[tex]\[ \frac{x^p}{x^p + x^q} \][/tex]
2. The second fraction is:
[tex]\[ \frac{1}{x^{p - q} + 1} \][/tex]
To simplify the sum of these fractions, we need a common denominator. The common denominator of the two fractions will be:
[tex]\[ (x^p + x^q)(x^{p - q} + 1) \][/tex]
Now, rewrite each fraction with this common denominator.
For the first fraction:
[tex]\[ \frac{x^p (x^{p - q} + 1)}{(x^p + x^q)(x^{p - q} + 1)} \][/tex]
For the second fraction:
[tex]\[ \frac{x^p + x^q}{(x^p + x^q)(x^{p - q} + 1)} \][/tex]
Now, let's add these fractions:
[tex]\[ \frac{x^p (x^{p - q} + 1) + x^p + x^q}{(x^p + x^q)(x^{p - q} + 1)} \][/tex]
Combine the terms in the numerator:
[tex]\[ x^p(x^{p - q} + 1) + x^p + x^q \][/tex]
Simplify the numerator further:
[tex]\[ x^p \cdot x^{p - q} + x^p \cdot 1 + x^p + x^q \][/tex]
Combine like terms:
[tex]\[ x^{2p - q} + 2x^p + x^q \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ \frac{x^{2p - q} + 2x^p + x^q}{(x^p + x^q)(x^{p - q} + 1)} \][/tex]
So the final result is:
[tex]\[ \frac{x^p}{x^p + x^q} + \frac{1}{x^{p - q} + 1} = \frac{x^{2p - q} + 2x^p + x^q}{(x^p + x^q)(x^{p - q} + 1)} \][/tex]
