\( \frac{\left(2^{3}\right)^{2} ; 2^{2}}{2^{4}} \quad \) 6) \( \frac{4^{4}}{4^{2} \cdot\left(4^{3}\right)^{2}} \)

Let's break this down! To simplify \( \frac{\left(2^{3}\right)^{2} ; 2^{2}}{2^{4}} \), first calculate \( (2^3)^2 = 2^{6} \). So, the expression becomes \( \frac{2^{6} \cdot 2^{2}}{2^{4}} \). Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we have \( 2^{6+2} = 2^{8} \). Now, simplifying \( \frac{2^{8}}{2^{4}} \) gives us \( 2^{8-4} = 2^{4} = 16 \). Now for \( \frac{4^{4}}{4^{2} \cdot\left(4^{3}\right)^{2}} \), we start with \( (4^3)^2 = 4^{6} \). The expression becomes \( \frac{4^{4}}{4^{2} \cdot 4^{6}} \). This combines in the denominator: \( 4^{2+6} = 4^{8} \). Now simplifying \( \frac{4^{4}}{4^{8}} = 4^{4-8} = 4^{-4} = \frac{1}{4^{4}} = \frac{1}{256} \). So, the final results are \( 16 \) and \( \frac{1}{256} \)!