\( \frac{4 d^{6}}{16 d^{2}}= \) \( \left(5^{4}\right)^{x}=5^{20} \). What is \( x ? \)

To simplify \( \frac{4 d^{6}}{16 d^{2}} \), start by reducing the coefficients and the powers of \( d \). The coefficient \( \frac{4}{16} \) simplifies to \( \frac{1}{4} \), and using the property of exponents \( \frac{d^{6}}{d^{2}} = d^{6-2} = d^{4} \). Thus, \( \frac{4 d^{6}}{16 d^{2}} = \frac{1}{4} d^{4} \). For the equation \( (5^{4})^{x} = 5^{20} \), use the property of exponents that states \( (a^{m})^{n} = a^{m \cdot n} \). So, \( (5^{4})^{x} = 5^{4x} \). Setting the exponents equal to each other gives you \( 4x = 20 \). Solving for \( x \) yields \( x = 5 \). Therefore, the simplified expression is \( \frac{1}{4} d^{4} \) and \( x = 5 \).