3) \( \frac{4^{2} \cdot 4^{4}}{4^{3}} \) 5) \( \frac{\left(2^{3}\right)^{2} \cdot 2^{2}}{2^{4}} \)

To simplify the first expression \( \frac{4^{2} \cdot 4^{4}}{4^{3}} \), you can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). Thus, you combine the numerators: \( 4^{2 + 4} = 4^{6} \). Now, the expression is \( \frac{4^{6}}{4^{3}} \). Again applying the property for division, \( \frac{a^m}{a^n} = a^{m-n} \): \( 4^{6-3} = 4^{3} \). So, the simplified result is \( 4^{3} = 64 \). For the second expression \( \frac{\left(2^{3}\right)^{2} \cdot 2^{2}}{2^{4}} \), utilize the exponent multiplication rule. First, simplify \( \left(2^{3}\right)^{2} = 2^{6} \) (because \( (a^m)^n = a^{mn} \)). The expression now looks like \( \frac{2^{6} \cdot 2^{2}}{2^{4}} \). Combining the top part, \( 2^{6 + 2} = 2^{8} \), leads to \( \frac{2^{8}}{2^{4}} \). Finally, applying the division rule gives us \( 2^{8-4} = 2^{4} \), which simplifies to \( 16 \).