Part 2)

a) Consider the 1 card to be 2^0.  Then there are 26 powers of 2 in the deck, half of them odd and half of them even.  If we pick three of them, their product will be a power of 2.  Half of those products will be an even power of two and thus a perfect square.  (This occurs when 1 or 3 of the three cards are an even power)  The probability of getting three cards that are powers of two are (26/52)*(25/51)*(24/50) = 6/51.  Half the time (with probability 3/51) this is a perfect square.

b) Similarly, the chance is 3/51 that the product is an even power of 3.

   

c) Or the 3 cards could be two powers of 2 (including 2^0) and an even power of 3 (excluding 3^0).  The probability of picking two powers of 2 and an even power of 3 are (26/52)*(25/51)*(12/50)*3 (because there are only 12 even powers of 3 and it could be picked first, second or third).  This equals 9/51.  But only half of these are a perfect square, because half of product of the powers of two have an even exponent (if both odd or both even).  So the probability of a non-zero perfect square this way is 4.5/51

d) Or the 3 cards could be two powers of 3 (including 3^0) and an even power of 2 (excluding 2^0).  The probability of this happenning and being a perfect square is also 4.5/51

   

   These four are mutually exclusive, and these are the only four ways to get a non-zero perfect square, so the probability of a non-zero perfect square is 3/51 + 3/51 + 4.5/51 + 4.5/51 = 15/51 = 5/17.