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Efectué y simplifique las siguientes expresiones utilizando la potenciación o 
la radicación.

I see you've uploaded an image with several mathematical expressions. Let's go through them step by step, simplifying each expression. Here's the breakdown:

1. **Expression 1:**
   \[
   \frac{x - 4(x - 2)}{x - 4}
   \]
   Simplify the numerator:
   \[
   x - 4(x - 2) = x - 4x + 8 = -3x + 8
   \]
   The expression becomes:
   \[
   \frac{-3x + 8}{x - 4}
   \]

2. **Expression 2:**
   \[
   \frac{3x - 4}{2x} - \frac{5x - 10}{4x^2}
   \]
   Simplify the second fraction:
   \[
   \frac{5x - 10}{4x^2} = \frac{5(x - 2)}{4x^2}
   \]
   Combine both terms using a common denominator \(4x^2\):
   \[
   \frac{(3x - 4)(2)}{4x^2} - \frac{5(x - 2)}{4x^2}
   \]
   Simplify further:
   \[
   \frac{6x - 8 - 5x + 10}{4x^2} = \frac{x + 2}{4x^2}
   \]

3. **Expression 3:**
   \[
   \frac{\sqrt{9x^2} \sqrt{25x^4}}{5x^8}
   \]
   Simplify the square roots:
   \[
   \sqrt{9x^2} = 3x, \quad \sqrt{25x^4} = 5x^2
   \]
   Thus, the expression becomes:
   \[
   \frac{3x \cdot 5x^2}{5x^8} = \frac{15x^3}{5x^8} = \frac{3}{x^5}
   \]

4. **Expression 4:**
   \[
   \frac{\sqrt{24x^4}}{\sqrt{2x^8}}
   \]
   Simplify the square roots:
   \[
   \sqrt{24x^4} = 2x^2\sqrt{6}, \quad \sqrt{2x^8} = \sqrt{2}x^4
   \]
   Thus, the expression becomes:
   \[
   \frac{2x^2\sqrt{6}}{\sqrt{2}x^4} = \frac{2\sqrt{6}}{\sqrt{2}x^2} = \frac{2\sqrt{3}}{x^2}
   \]

5. **Expression 5:**
   \[
   (x + 8)^{-\frac{9}{2}} (x + 8) (x + 8)^{\frac{3}{2}}
   \]
   Combine the exponents:
   \[
   (x + 8)^{-\frac{9}{2} + 1 + \frac{3}{2}} = (x + 8)^{-\frac{9}{2} + \frac{5}{2} + \frac{3}{2}} = (x + 8)^{-\frac{1}{2}}
   \]
   Final expression:
   \[
   \frac{1}{\sqrt{x + 8}}
   \]

6. **Expression 6:**
   \[
   \frac{2x - 6}{4x^2 - 9} + \frac{x - 8}{4x^2 + 9}
   \]
   Factor the first denominator:
   \[
   4x^2 - 9 = (2x - 3)(2x + 3)
   \]
   Combine the terms over a common denominator:
   \[
   \frac{2(x - 3)}{(2x - 3)(2x + 3)} + \frac{x - 8}{4x^2 + 9}
   \]
   Further simplification depends on the final combination, but it requires factoring.

7. **Expression 7:**
   \[
   \sqrt{20} - 5\sqrt{24} + 8\sqrt{50} - \sqrt{18}
   \]
   Simplify the square roots:
   \[
   \sqrt{20} = 2\sqrt{5}, \quad \sqrt{24} = 2\sqrt{6}, \quad \sqrt{50} = 5\sqrt{2}, \quad \sqrt{18} = 3\sqrt{2}
   \]
   Thus, the expression becomes:
   \[
   2\sqrt{5} - 10\sqrt{6} + 40\sqrt{2} - 3\sqrt{2} = 2\sqrt{5} - 10\sqrt{6} + 37\sqrt{2}
   \]

8. **Expression 8:**
   \[
   \frac{6x^4 \sqrt{9}}{6x^4}
   \]
   Simplify:
   \[
   \sqrt{9} = 3
   \]
   Thus, the expression becomes:
   \[
   \frac{6x^4 \cdot 3}{6x^4} = 3
   \]

If you would like more details on any specific step or have any questions, feel free to ask!

Here are 5 follow-up questions to expand your understanding:
1. How can common denominators be used to simplify rational expressions?
2. What strategies can help when simplifying complex square roots?
3. How do exponent rules assist in combining terms with exponents?
4. What does it mean to factor denominators in rational expressions?
5. How can roots of non-perfect squares be handled in expressions?

**Tip:** Always look for common factors and simplifications early on, it saves time when dealing with complex expressions.

Efectué y simplifique las siguientes expresiones utilizando la potenciación o la radicación.

Learn how to simplify complex mathematical expressions involving radicals, exponents, and rational terms. This problem covers the use of exponent rules, factoring in algebra, and radical simplifications. Suitable for students in Grades 10-12.

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