A question – 64 is a cube and a square; name another such number

NO CALCULATORS

 Should this question be in the “no calculators” section of a 7^th grade paper?

If Yes, it is a challenge to Budding Mathematician, who we can call BM. Best marks are usually won from a combination of discipline and simplicity.

            64  = 8^2 and 4^3
but the key to finding 
another qualifier 
is recognizing that 
            64 = 2^6

The rule is no calculators, so if BM actually gets to

           3^6 = 729,
it is not a number that is likely to jump to mind as a perfect square. 

Prime factors could be used to check, and once 3*3*81 is reached, confidence would creep in.

           4^6 = 4096; if BM remembered repeated doubling of 1 and reaching 1024,
it is just another two doublings,
but how many doublings was that altogether? 

Determination would suffer repeating the process, 

and again the confidence would allow some satisfaction.

Does this work have to be so hard? No!

1^2 is so obvious, that there is

no hesitation with 10^2 and 100^2

Finally BM sees that
1,000,000 = 10^6
is the simple answer,
so allows a big grin to spread over the BM face.
This is a first, and an empowering one. 

Never before had the number
1,000,000; 

big as it is, 

been the simple answer .