Alright, let's break down the given expression step by step:
Given expression:
[tex]\[
\frac{x^p}{x^p + x^q} + \frac{1}{x^{p-q} + 1}
\][/tex]
Let's denote the terms individually for clarity:
- Term 1: [tex]\(\frac{x^p}{x^p + x^q}\)[/tex]
- Term 2: [tex]\(\frac{1}{x^{p-q} + 1}\)[/tex]
We'll add these two terms together.
### Step-by-Step Simplification:
Step 1: Simplify each term separately.
- Term 1: [tex]\(\frac{x^p}{x^p + x^q}\)[/tex]
This term is already in its simplest form since it represents a ratio of two expressions.
- Term 2: [tex]\(\frac{1}{x^{p-q} + 1}\)[/tex]
This term is also in its simplest form.
Step 2: Add the two terms together with a common denominator.
To add these two terms, we need a common denominator. The common denominator for these terms would be:
[tex]\[
(x^p + x^q)(x^{p-q} + 1)
\][/tex]
Rewrite each fraction with this common denominator:
- Rewrite Term 1 with the common denominator:
[tex]\[
\frac{x^p}{x^p + x^q} = \frac{x^p (x^{p-q} + 1)}{(x^p + x^q)(x^{p-q} + 1)}
\][/tex]
- Rewrite Term 2 with the common denominator:
[tex]\[
\frac{1}{x^{p-q} + 1} = \frac{x^p + x^q}{(x^p + x^q)(x^{p-q} + 1)}
\][/tex]
Step 3: Combine the two rewritten terms.
Now we add the rewritten terms:
[tex]\[
\frac{x^p (x^{p-q} + 1)}{(x^p + x^q)(x^{p-q} + 1)} + \frac{x^p + x^q}{(x^p + x^q)(x^{p-q} + 1)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{x^p (x^{p-q} + 1) + (x^p + x^q)}{(x^p + x^q)(x^{p-q} + 1)}
\][/tex]
Step 4: Simplify the combined numerator.
Combine the terms in the numerator:
[tex]\[
x^p (x^{p-q} + 1) + x^p + x^q
\][/tex]
Distribute [tex]\(x^p\)[/tex]:
[tex]\[
x^p x^{p-q} + x^p + x^p + x^q = x^{2p-q} + x^p + x^p + x^q
\][/tex]
Finally, combine like terms:
[tex]\[
x^{2p-q} + 2x^p + x^q
\][/tex]
Therefore, the simplified expression becomes:
[tex]\[
\frac{x^{2p-q} + 2x^p + x^q}{(x^p + x^q)(x^{p-q} + 1)}
\][/tex]
The overall simplified form for the given expression is:
[tex]\[
\frac{x^p (x^{p-q} + 1) + x^p + x^q}{(x^p + x^q)(x^{p-q} + 1)}
\][/tex]
