We are tasked with simplifying the following expression:

$8\log_4(2) + \frac{\log_4(64)}{3 \log_4(4)}$

Step 1: Break down logarithmic terms.

We'll first simplify the logarithmic expressions individually.

1. $\log_4(2)$:
   We can rewrite $2$ as $4^{1/2}$, so: $\log_4(2) = \log_4(4^{1/2}) = \frac{1}{2}$

2. $\log_4(64)$:
   Since $64 = 4^{3/2}$, we have: $\log_4(64) = \log_4(4^{3/2}) = \frac{3}{2}$

3. $\log_4(4)$:
   Since $4 = 4^1$, we know: $\log_4(4) = 1$

Step 2: Substitute and simplify.

Now, substitute these values into the original expression:

$8\left(\frac{1}{2}\right) + \frac{\frac{3}{2}}{3 \times 1}$

Simplifying each part: $8\left(\frac{1}{2}\right) = 4$ $\frac{\frac{3}{2}}{3} = \frac{3}{2} \times \frac{1}{3} = \frac{1}{2}$

Thus, the entire expression becomes: $4 + \frac{1}{2} = 4.5$

Final Answer:

The simplified value of the expression is:

$\boxed{4.5}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the result change if the base of the logarithms was different?
2. What is the general process for simplifying logarithmic expressions with different bases?
3. How can you express logarithmic identities in terms of natural logarithms?
4. What is the logarithmic change of base formula, and how does it apply here?
5. How do you graph a function involving logarithms with different bases?

Tip: When working with logarithmic expressions, always try to express numbers in terms of powers of the base to simplify them effectively.