Data Sufficiency

For each of the following, could the answer
be an integer if x is an integer greater than 1?
a) x10 + x-10 =
b) x1/6 + x1/2 =

for option b why it will be an integer?

Notice that the question is not "will the answer always be an integer?"

The question is "could the answer be an integer?"

Any number with an integer sixth root will also have an integer square root, as the square root is the cube of the sixth root.

So for any number such that the sixth root is an integer, x¹/� + x¹/² will be an integer.

Another way to look at is that for any number x such that x = an integer raised to the sixth power, x¹/� + x¹/² will be an integer.

Another thing worth pointing out here is that

(x¹/�)³ = (x¹/²)

So if x¹/� is an integer, x¹/² will be as well.