Perfect Square/Cube

 

Which number (after 64) is the next number that is BOTH a perfect square and a perfect cube?  64 = 8 squared and 64 = 4 cubed  

 

The answer is :729

Here is my work:

The next fifth cube number is 125 because 5x5x5=125

But it can't be a perfect square and a perfect cube although 125=5 cubed.

The next sixth cube number is 216 because 6x6x6=216

But it can't be a perfect square and a perfect cube although 216=6 cubed. 

The next seventh cube number is 343 because 7x7x7=343

But it can't be a perfect square and a perfect cube although 343=7 cubed

The next eighth cube number is 512 because 8x8x8=512

But it can't be a perfect square and a perfect cube although 512=8 cubed

The next ninth cube number is 729 because 9x9x9=729

But it CAN be a perfect square and a perfect cube although 729=9 cubed

729= 27 squared and 729=9 cubed.