https://file.aiquickdraw.com/mathbot/file/17261230021572ce83csx.jpg

The expression in the image is \((x + \frac{2}{x})^{10}\). 

To expand this expression, we would typically use the **binomial theorem**, which states that:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

Here, \(a = x\), \(b = \frac{2}{x}\), and \(n = 10\).

Thus, we can write the expanded form as:

\[
(x + \frac{2}{x})^{10} = \sum_{k=0}^{10} \binom{10}{k} x^{10-k} \left(\frac{2}{x}\right)^k
\]

This leads to the following general term in the expansion:

\[
\binom{10}{k} x^{10-k} \cdot \frac{2^k}{x^k} = \binom{10}{k} 2^k x^{10-2k}
\]

The powers of \(x\) will depend on the specific value of \(k\), and you will get terms where \(x\) is raised to different powers, which can range from \(x^{10}\) to \(x^{-10}\).

Would you like a fully expanded version of this expression or any further details? Let me know if you'd like to dive deeper into any specific part!

Here are some related questions you might consider:

1. How is the binomial theorem used to expand any power of a binomial expression?
2. What is the role of binomial coefficients in polynomial expansions?
3. How does the combination formula \(\binom{n}{k}\) work?
4. What are some examples of applying the binomial theorem to more complex expressions?
5. How does the power of \(x\) change across the expansion terms?

**Tip:** When dealing with binomial expansions involving rational expressions, keep track of how exponents of variables interact (e.g., positive and negative powers of \(x\)).

Learn how to expand the expression (x + 2/x)^10 using the Binomial Theorem. This step-by-step guide covers the application of the Binomial Theorem to rational expressions, suitable for students in Grades 10-12.

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