Calculator Fun

Once, I was playing with my calculator at school during the break. I was doing many things with the number 16, when I decided to calculate the square root of it. The result, of course, was 4. However, I pressed, by accident, the square root button again, and the result that was being shown was 2. I then thought of what had just happened: I had a number, then I calculated the square root of it, and the square root of the result was an integer, which was not a perfect square. In other words, the result of the square root of a perfect square (16) was also a perfect square (4). Let the initial number be xx, and the perfect square results be x1,x2,x3,...,xmx_{1}, x_{2}, x_{3}, ... ,x_{m}, where xmx_{m} is the last possible result, which is not a perfect square. As an explicit example, I used x=16x = 16 that day at school, so mm was 22 and xm=2x_{m} = 2. Consider that nn and kk are positive integers. In order to maximize the value of mm, xx should be of the form:

Note: the exponent of xmx_m must be minimized.

Image credit: Like Cool.
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