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.