In a radioactive decay process, the activity is defined as \( A = \dfrac{-dN}{dt} \), where \( N(t) \) is the number of radioactive nuclei at time \( t \). Two radioactive sources, \( S_1 \) and \( S_2 \) have same activity at time \( t = 0 \). At a later time, the activities of \( S_1 \) and \( S_2 \) are \( A_1 \) and \( A_2 \), respectively. When \( S_1 \) and \( S_2 \) have just completed their \( 3^{\text{rd}} \) and \( 7^{\text{th}} \) half-lives, respectively, the ratio \( \dfrac{A_1}{A_2} \) is __________.
