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

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About mvtrinh

Retired high school math teacher.
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