## Perfect Squares and Perfect Cubes

List all integers less than $20,\!000$ that are both perfect squares and perfect cubes.
Source: NCTM Mathematics Teacher, January 2006

Solution
$9=3^2$ is a perfect square but not a perfect cube. On the other hand, $8=2^3$ is a perfect cube but not a perfect square. For an integer to be both perfect square and perfect cube, the exponent must be a multiple of $2$ and $3$, that is, the integer must be a sixth power.
$0^6=0$
$1^6=1$
$2^6=64$
$3^6=729$
$4^6=4096$
$5^6=15625$
$6^6=46656>20000$
All integers less than $20,\!000$ that are both perfect squares and perfect cubes
$0,1,64,729,4096,15625$

Answer: $0,1,64,729,4096,15625$