Number with a sixth root will have an integer square root

[gayatri.k]

Hello!

Does anyone have a proof / explanation / solution for the following question?

"Any number with an integer sixth root will have an integer square root. Why?"

This question can be found on MGMATs EIVs flash cards, card number 2.

Thank you!

Gayatri

[RonPurewal]

consider what happens when you take the sixth root and cube it.

[RonPurewal]

if that hint ^^ is too cryptic, then here are some additional hints:

* if you are "a number with an integer sixth root", then you are 1^6, or 2^6, or 3^6, or, in general, integer^6... and that integer is your sixth root.

* in that case, what is your square root?
it's integer^3.
if you don't see why, just play around with some of the specific numbers listed above. you'll see how it works.

--

most compact explanation possible:
(sixth root of x)^3 = √x

[gayatri.k]

That makes sense! Thank you Ron.

[RonPurewal]

you're welcome.

[KakakkdeK400]

Thank you for the informative posts.

[RonPurewal]

no problem.