Use properties of logarithms to evaluate without using a calculator. \[ 8 \log _{4}(2)+\frac{\log _{4}(64)}{3 \log _{4}(4)} \]

To evaluate the expression, we can first simplify each part using the properties of logarithms. The first part is \(8 \log_{4}(2)\). We know that \( \log_{4}(2) = \frac{1}{2} \) because \(2\) is \(4^{1/2}\). Therefore, \[ 8 \log_{4}(2) = 8 \cdot \frac{1}{2} = 4. \] The second part is \( \frac{\log_{4}(64)}{3 \log_{4}(4)} \). We first simplify \( \log_{4}(64) \). Noting that \(64 = 4^3\), we have \[ \log_{4}(64) = 3. \] Also, since \( \log_{4}(4) = 1\), we find that \[ 3 \log_{4}(4) = 3 \cdot 1 = 3. \] Thus, \[ \frac{\log_{4}(64)}{3 \log_{4}(4)} = \frac{3}{3} = 1. \] Now combining both parts: \[ 8 \log_{4}(2) + \frac{\log_{4}(64)}{3 \log_{4}(4)} = 4 + 1 = 5. \] So, the final answer is \[ \boxed{5}. \]