A perfect cube is an integer whose cube root is an integer. For example, 27, 64 and 125 are perfect cubes. If p and q are perfect cubes, which of the following will not necessarily be a perfect cube?
a) 8p b) pq c) pq + 27 d) -p e) (p - q)6

• c) pq + 27
  e) (p - q)6
  
  These two terms may not be perfect cubes.
  If last term is (p-q)^6 , then only [ pq + 27] is such term.
• 13 years agoHelpfull: Yes(5) No(9)
• 
  
  A perfect cube will have prime factors that are in groups of 3; for example 125 has the prime factors 5 x 5 x 5 , and 64 x 125 will also be a cube because its factors will be 4 x 4 x 4 x 5 x 5 x 5
  
  Consider the answer choices in turn.
  
  8 is the cube of 2, and p is a cube, and so the product will also be a cube.
  
  pq will also be a cube as shown above.
  
  pq is a cube and so is 27, but their sum need not be a cube. Consider the case where p =1 and q = 8, the sum of pq and 27 will be 35 which has factors 5 x 7 and is not a cube.
  
  -p will be a cube.
  
  Since the difference between p and q is raised to the power of 6, this expression will be a cube (with cube root = difference squared).
• 9 years agoHelpfull: Yes(2) No(0)
• perfect cube multiplied or divided by another perfect cube yields a perfect cube. Therefore,
  
  (c) and (e) are not NECESSARILY perfect cubes.
• 10 years agoHelpfull: Yes(1) No(1)