https://file.aiquickdraw.com/mathbot/file/1727075215843zovvvoho.jpg

The given expression in the image is:

\[
\frac{1}{1 + a^{n - m}} + \frac{1}{1 + a^{m - n}}
\]

To simplify this expression, follow these steps:

### Step 1: Identify symmetry
The terms \(a^{n - m}\) and \(a^{m - n}\) are related by the property \(a^{m - n} = \frac{1}{a^{n - m}}\). This allows us to rewrite the second term as:

\[
\frac{1}{1 + a^{m - n}} = \frac{1}{1 + \frac{1}{a^{n - m}}}
\]

### Step 2: Combine the terms
Now, simplify \(\frac{1}{1 + \frac{1}{a^{n - m}}}\):

\[
\frac{1}{1 + \frac{1}{a^{n - m}}} = \frac{a^{n - m}}{a^{n - m} + 1}
\]

So, the original expression becomes:

\[
\frac{1}{1 + a^{n - m}} + \frac{a^{n - m}}{a^{n - m} + 1}
\]

### Step 3: Add the two fractions
Both fractions now have the same denominator, \(1 + a^{n - m}\), so we can directly add them:

\[
\frac{1}{1 + a^{n - m}} + \frac{a^{n - m}}{1 + a^{n - m}} = \frac{1 + a^{n - m}}{1 + a^{n - m}} = 1
\]

### Final Answer:
\[
\frac{1}{1 + a^{n - m}} + \frac{1}{1 + a^{m - n}} = 1
\]

Would you like more details on this solution or have any questions?

Here are 5 related questions to practice:
1. Simplify \(\frac{1}{1 + a^{m - n}} + \frac{1}{1 + a^{2m - 2n}}\).
2. What is the value of \(\frac{1}{1 + b^{p - q}} + \frac{1}{1 + b^{q - p}}\)?
3. Solve \(\frac{1}{1 + x^2} + \frac{1}{1 + \frac{1}{x^2}}\).
4. What is the general form of \(\frac{1}{1 + x} + \frac{1}{1 + \frac{1}{x}}\)?
5. Simplify \(\frac{1}{1 + e^{x}} + \frac{1}{1 + e^{-x}}\).

**Tip:** Always check for symmetry or inverses when working with powers or fractions, as they can often simplify the problem quickly.

1 / (1 + a^(n - m)) + 1 / (1 + a^(m - n))

Learn how to simplify the expression 1 / (1 + a^(n - m)) + 1 / (1 + a^(m - n)) using exponent rules and fraction simplifications. This problem involves symmetry in expressions and is ideal for algebra students in Grades 9-12.

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