Express the following fraction in simplest form using only positive exponents. \[ \frac{\left(4 u^{2}\right)^{4}}{12 u^{5}} \]

To simplify the fraction \(\frac{\left(4 u^{2}\right)^{4}}{12 u^{5}}\), we start with the numerator: \[ (4 u^2)^4 = 4^4 (u^2)^4 = 256 u^8 \] Now, substituting back into the fraction gives us: \[ \frac{256 u^8}{12 u^5} \] Next, we can simplify the coefficients \(256\) and \(12\). The greatest common divisor of \(256\) and \(12\) is \(4\): \[ \frac{256}{12} = \frac{256 \div 4}{12 \div 4} = \frac{64}{3} \] So, our fraction becomes: \[ \frac{64 u^8}{3 u^5} \] Now, we simplify the variable part by using the laws of exponents: \[ \frac{u^8}{u^5} = u^{8-5} = u^3 \] Thus, our simplified expression is: \[ \frac{64 u^3}{3} \] This means the fraction in simplest form using only positive exponents is: \[ \frac{64 u^3}{3} \]