2k is a perfect square, 3k is a perfect cube, and 5k is a perfect 4th power. What is the sum of the exponents in the prime factorization of the smallest such positive integer k ?

The immage below is simmilar question...The different between the two question > "5k is a perfect 4th power." and your question "5k is a perfect 5th power."

Exactly my point. Changing 5th to 4th makes it impossible. I can show it's impossible but I'll leave you to see it for yourself. I've solved and posted a solution for 5th already.