Question

The number of integers n such that 1less than or equal to n less than or equal to 200 and nⁿ is a perfect square is ?

Answer 1

n^n is perfect when n is even,
integers satisfying this condition belong to the set A={2,4,6,⋯198,200} So, the number of possibilities are :100.

(2) You have to include those n's that are perfect squares as well. In, this case, n=k^2. So, n^n=(k2^)^(k^2)which is a perfect square. Counting all odd perfect squares, to prevent double counting of even ones from (1). The integers belong to the set B={1,9,25,⋯169} So, the number of possibilities are: 7.

We thus have a total of 107 integers satisfying the condition.