The tutor investigates numbers that have both whole number square and cube roots.

Numbers that have both a whole number square root and cube root are rare. 64 is one: 64^(1/2)=8, while 64^(1/3)=4. 729 is next: 729^(1/2)=27, while 729^(1/3)=9.

The reason 64 and 729 are both perfect squares as well as perfect cubes is that they are both perfect sixth powers: 64=2^6; 729=3^6. Therefore, they can be arranged as (a^3)^2 as well as (a^2)^3. The square root is then a^3, while the cube root is a^2.

I’ll be talking more about perfect roots in future posts:)

Source:

Travers, Kenneth J., et al. Using Advanced Algebra. Toronto: Doubleday Canada, 1977.