The number of positive integers from 1 to 106 (both inclusive), which are perfect squares, perfect cubes or perfect 4th powers is

To solve this problem, we need to find the number of perfect squares, perfect cubes, and perfect fourth powers between 1 and 106.

1. Perfect Squares: The perfect squares between 1 and 106 are the squares of the integers from 1 to 10 (since 11^2 = 121 which is greater than 106). So, there are 10 perfect squares.

2. Perfect Cubes: The perfect cubes between 1 and 106 are the cubes of the integers from 1 to 4 (since 5^3 = 125 which is greater than 106). So, there are 4 perfect cubes.

3. Perfect Fourth Powers: The perfect fourth powers between 1 and 106 are the fourth powers of the integers from 1 to 3 (since 4^4 = 256 which is greater than 106). So, there are 3 perfect fourth powers.

However, note that some numbers may be counted more than once. For example, 1 is a perfect square, a perfect cube, and a perfect fourth power. Similarly, 16 is both a perfect square and a perfect fourth power. So, we need to subtract the duplicates.

The duplicates are the numbers which are both perfect squares and perfect fourth powers. These are 1^4, 2^4, and 3^4, so there are 3 duplicates.

So, the total number of distinct positive integers between 1 and 106 that are perfect squares, perfect cubes, or perfect fourth powers is 10 (perfect squares) + 4 (perfect cubes) + 3 (perfect fourth powers) - 3 (duplicates) = 14.