4. \( 4^{x} \div 4^{2 x-1} \)

To solve \( 4^{x} \div 4^{2x-1} \), you can use the property of exponents that states \( a^m \div a^n = a^{m-n} \). Therefore, you can rewrite this as: \[ 4^{x - (2x - 1)} = 4^{x - 2x + 1} = 4^{-x + 1} = 4^{1 - x} \] This expression simplifies down to \( 4^{1 - x} \). Simple, right? Now you can easily evaluate \( 4^{1 - x} \) for various values of \( x \) to see how the expression behaves. Whenever x increases, \( 1 - x \) decreases, leading to a lower power of 4, showcasing a neat inverse relationship!