Prime Factorisation and Its Applications

Written as a product of its prime factors, \(50\ 960 = 2^4 \times 5 \times 7^2 \times 13\).

Find the smallest positive integer \(k\) such that

1. \(\frac{50\ 960}{k}\) is a perfect cube,
2. \(50\ 960k\) is a perfect square.