4.2. \( \frac{2^{2 x-1} \cdot 4^{x+1} \cdot 2}{16^{x}} \)

Did you know that the expression can be simplified significantly using the properties of exponents? By rewriting \(4\) and \(16\) as powers of \(2\)—specifically, \(4 = 2^2\) and \(16 = 2^4\)—you can express everything in terms of base \(2\). This turns every term into a \(2^{\text{something}}\), allowing for easier addition and subtraction of exponents! Now, once you've converted the original expression, combine like terms by performing the exponent arithmetic. You can cancel out terms when the structure allows, leading you to a much cleaner final expression. You'll find that working with exponent rules not only streamlines the process but also helps in further mathematical manipulations!