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To solve the equation \( 2^{x-4} = 64 \sqrt{2} \), we can follow these steps:

1. **Express both sides with the same base:**

   We know that:
   \[
   64 = 2^6 \quad \text{and} \quad \sqrt{2} = 2^{1/2}
   \]

   Therefore,
   \[
   64 \sqrt{2} = 2^6 \cdot 2^{1/2} = 2^{6.5} = 2^{13/2}
   \]

2. **Rewrite the equation with the common base:**

   \[
   2^{x-4} = 2^{13/2}
   \]

3. **Equate the exponents:**

   Since the bases are the same, we can set the exponents equal to each other:
   \[
   x - 4 = \frac{13}{2}
   \]

4. **Solve for \( x \):**

   Add 4 to both sides:
   \[
   x = \frac{13}{2} + 4
   \]

   Convert 4 to a fraction with the same denominator:
   \[
   4 = \frac{8}{2}
   \]

   Now add the fractions:
   \[
   x = \frac{13}{2} + \frac{8}{2} = \frac{21}{2}
   \]

So, the value of \( x \) that satisfies the equation is \( \frac{21}{2} \).

5. **Convert the answer into the form given in the options:**

   \[
   \frac{21}{2} = 10 \frac{1}{2}
   \]

Therefore, the correct answer is:
\[ \boxed{\text{a. } 10 \frac{1}{2}} \]

Would you like more details or have any questions? Here are some related questions to consider:

1. How would you solve the equation \(3^{x-2} = 27 \sqrt{3}\)?
2. What are the steps to express 128 in terms of base 2?
3. How do you convert a decimal to a fraction?
4. Can you explain the properties of exponents used in solving equations like these?
5. What is the significance of expressing numbers with the same base in solving exponential equations?

**Tip:** When solving exponential equations, expressing all terms with a common base can greatly simplify the problem.

Nilai x yang memenuhi persamaan 2^{x-4} = 64 √2 adalah .

Solve the exponential equation 2^{x-4} = 64√2 step-by-step, using properties of exponents and simplifying the terms. The solution requires expressing terms with the same base and solving for x. Ideal for students in Grades 10-12.

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