To solve the expression [tex]\(\frac{1}{1 + a^{n-m}} + \frac{1}{1 + a^{m-n}}\)[/tex], we need to carefully simplify the terms involved. Follow these steps:
1. Identify the terms and their properties:
- The exponents [tex]\(n - m\)[/tex] and [tex]\(m - n\)[/tex] are just negatives of each other: [tex]\(m - n = -(n - m)\)[/tex].
2. Rewrite the second fraction using the property of exponents:
- Notice that [tex]\(a^{m - n} = \left(a^{n - m}\right)^{-1} = \frac{1}{a^{n - m}}\)[/tex].
3. Simplify each fraction:
- The first fraction is [tex]\(\frac{1}{1 + a^{n - m}}\)[/tex], so it remains as is.
- Rewrite the second fraction using our observation: [tex]\(\frac{1}{1 + a^{m - n}} = \frac{1}{1 + \frac{1}{a^{n - m}}}\)[/tex].
4. Simplify the denominator of the second fraction:
- Combine the terms in the denominator: [tex]\(1 + \frac{1}{a^{n - m}} = \frac{a^{n - m} + 1}{a^{n - m}}\)[/tex].
5. Invert the second fraction:
- [tex]\(\frac{1}{1 + \frac{1}{a^{n - m}}} = \frac{a^{n - m}}{a^{n - m} + 1}\)[/tex].
6. Combine the terms:
- Now we have [tex]\(\frac{1}{1 + a^{n - m}} + \frac{a^{n - m}}{a^{n - m} + 1}\)[/tex].
7. Identify the common structure:
- Let [tex]\(x = a^{n - m}\)[/tex]. The expression becomes [tex]\(\frac{1}{1 + x} + \frac{x}{1 + x}\)[/tex].
8. Add the fractions:
- Since the denominators are the same, [tex]\(\frac{1 + x}{1 + x} = 1\)[/tex].
Thus, the final simplified expression is:
[tex]\[
\frac{1}{1 + a^{n-m}} + \frac{1}{1 + a^{m-n}} = 1.
\][/tex]
