The expression you've provided is:

\[
(3 - x) \cdot \sqrt{\frac{x^2 + 3}{(x - 3)^3}}
\]

### Let's simplify and analyze this step-by-step.

1. **Factor the term \( 3 - x \):**

   Notice that \( 3 - x = -(x - 3) \). So we can rewrite the expression as:

   \[
   -(x - 3) \cdot \sqrt{\frac{x^2 + 3}{(x - 3)^3}}
   \]

2. **Combine the terms involving \( (x - 3) \):**

   The expression now becomes:

   \[
   -\frac{(x - 3)}{\sqrt{(x - 3)^3}} \cdot \sqrt{x^2 + 3}
   \]

   Since \( \sqrt{(x - 3)^3} = (x - 3)^{3/2} \), we simplify the expression further:

   \[
   -\frac{(x - 3)}{(x - 3)^{3/2}} \cdot \sqrt{x^2 + 3}
   \]

3. **Simplify the powers of \( (x - 3) \):**

   The powers of \( (x - 3) \) simplify as:

   \[
   \frac{(x - 3)}{(x - 3)^{3/2}} = \frac{1}{(x - 3)^{1/2}} = \frac{1}{\sqrt{x - 3}}
   \]

   So the expression becomes:

   \[
   -\frac{\sqrt{x^2 + 3}}{\sqrt{x - 3}}
   \]

4. **Final simplified expression:**

   The final simplified form of the given expression is:

   \[
   -\sqrt{\frac{x^2 + 3}{x - 3}}
   \]

This is the simplified version of the expression.

Would you like further details or have any questions?

### Related Questions:
1. How can I rationalize the denominator in the final result?
2. What happens if \( x = 3 \)? Is the expression undefined?
3. Can this expression be integrated or differentiated? How would you approach it?
4. What if \( (x - 3) \) were squared instead of cubed? How would the process change?
5. How would this expression change if we swapped \( 3 - x \) for \( x - 3 \)?

### Tip:
Always check for values that make the denominator zero to identify potential points of discontinuity in an expression.

Learn how to simplify the algebraic expression (3 - x)√((x^2 + 3)/(x - 3)^3) with a detailed step-by-step breakdown. This problem involves key algebraic concepts such as exponent rules, square root properties, and expression simplification. Suitable for students in Grades 10-12.

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