$$  $$($$ \sqrt{\placeholder{}} $$$$ \sqrt{x-3} $$​+3)$$ ^2 $$

It looks like there is some formatting confusion in the expression provided. From what I understand, it seems you want to simplify or solve an expression involving a square root and an exponent. Let's rewrite it clearly:

\[ \left( \sqrt{x-3} + 3 \right)^2 \]

To simplify this expression, we can expand it using the binomial theorem. Specifically, for any $ a $ and $ b $:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

Here, $ a = \sqrt{x-3} $ and $ b = 3 $. Applying the binomial expansion:

\[ \left( \sqrt{x-3} + 3 \right)^2 = (\sqrt{x-3})^2 + 2 \cdot \sqrt{x-3} \cdot 3 + 3^2 \]

Now, simplify each term:

1. $(\sqrt{x-3})^2 = x - 3$
2. $2 \cdot \sqrt{x-3} \cdot 3 = 6\sqrt{x-3}$
3. $3^2 = 9$

Combine these terms:

\[ \left( \sqrt{x-3} + 3 \right)^2 = x - 3 + 6\sqrt{x-3} + 9 \]

Simplify the expression further:

\[ \left( \sqrt{x-3} + 3 \right)^2 = x + 6\sqrt{x-3} + 6 \]

So the simplified form is:

\[ \left( \sqrt{x-3} + 3 \right)^2 = x + 6\sqrt{x-3} + 6 \]

Do you want more details or have any questions about this solution?

Here are some related questions:

1. How do you expand expressions using the binomial theorem?
2. What is the process for simplifying square root expressions?
3. How do you combine like terms in polynomial expressions?
4. What are the properties of square roots in algebra?
5. How do you solve equations involving square roots?
6. What are common mistakes to avoid when expanding binomials?
7. Can you show an example of expanding a binomial with different terms?
8. What are the applications of binomial expansions in calculus?

**Tip:** Always verify each step when working with binomial expansions to ensure accuracy, especially when dealing with square roots.

Learn how to simplify the expression (sqrt{x-3} + 3)^2 using the binomial expansion method. This problem covers advanced algebraic concepts, including binomial theorem applications, exponents, and square roots. Suitable for advanced high school students.

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