Find the number of positive integers not exceeding 1000 that are either the square or the cube of an integer

38 is the correct answer!

There are 10 perfect cubes between 1 and 1000 (namely cubes of 1-10)

There are 31 perfect squares between 1 and 1000 (squares from 1-31)

Out of these, 3 perfect squares are also perfect cubes,

{1^2=1^3, 8^2=4^3, 27^2=9^3}.

By the law of inclusion/exclusion, the number of positive integers not exceeding 1000 that are either the square or the cube of an integer

=10+31-3

=38