https://file.aiquickdraw.com/mathbot/file/1725635377677axzguc0u.jpg

The expression in the image you've uploaded appears to be:

\[
(x + y) \left( x^{-1} + y^{-1} \right)^{-1}
\]

Let's break it down step by step:

1. **First Term**: \((x + y)\)
   - This is a simple addition of two variables, \(x\) and \(y\).

2. **Second Term**: \(\left( x^{-1} + y^{-1} \right)^{-1}\)
   - \(x^{-1}\) is the reciprocal of \(x\), i.e., \(\frac{1}{x}\).
   - \(y^{-1}\) is the reciprocal of \(y\), i.e., \(\frac{1}{y}\).
   - When we add these, we get: \(\frac{1}{x} + \frac{1}{y}\).
   - The reciprocal of this sum is \(\left( \frac{1}{x} + \frac{1}{y} \right)^{-1}\).

Now, simplifying further:
\[
\left( \frac{1}{x} + \frac{1}{y} \right) = \frac{x + y}{xy}
\]
So, the reciprocal of this is:
\[
\left( \frac{x + y}{xy} \right)^{-1} = \frac{xy}{x + y}
\]

Finally, multiplying the two parts together:
\[
(x + y) \times \frac{xy}{x + y}
\]
Since the \((x + y)\) cancels out, the result is:
\[
xy
\]

Thus, the simplified expression is \(xy\).

Let me know if you'd like more details or have any questions! Here are some related questions for deeper understanding:

1. How do you simplify more complex rational expressions involving sums and reciprocals?
2. What are some common properties of reciprocals that help in simplifications?
3. How does factoring play a role in simplifying algebraic expressions?
4. What is the difference between multiplying and adding terms with negative exponents?
5. How do we apply the distributive property when dealing with complex fractions?

**Tip:** Always look for opportunities to cancel terms early in an expression to make simplification faster and more efficient.

Learn how to simplify the rational expression (x + y)(x^{-1} + y^{-1})^{-1} step by step. This solution breaks down key algebraic concepts, including reciprocals and the distributive property, making it suitable for students in Grades 9-11.

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