4.3. Simplify \( \frac{16^{n} \times 2^{n+1}}{4^{n}} \)

To simplify the expression \( \frac{16^{n} \times 2^{n+1}}{4^{n}} \), we can start by rewriting the bases in terms of powers of 2: - \( 16^n = (2^4)^n = 2^{4n} \) - \( 4^n = (2^2)^n = 2^{2n} \) Now substituting these back into the expression gives us: \[ \frac{2^{4n} \times 2^{n+1}}{2^{2n}} \] Next, we can combine the terms in the numerator. Applying the property of exponents \( a^m \times a^n = a^{m+n} \), we get: \[ 2^{4n + n + 1} = 2^{5n + 1} \] Now, substituting this into the fraction gives us: \[ \frac{2^{5n + 1}}{2^{2n}} \] Using the division property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify it to: \[ 2^{(5n + 1) - 2n} = 2^{3n + 1} \] Thus, the simplified expression is: \[ \boxed{2^{3n + 1}} \]