# Calculator Fun

Number Theory Level 4

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 $$x$$, and the perfect square results be $$x_{1}, x_{2}, x_{3}, ... ,x_{m}$$, where $$x_{m}$$ is the last possible result, which is not a perfect square. As an explicit example, I used $$x = 16$$ that day at school, so $$m$$ was $$2$$ and $$x_{m} = 2$$. Consider that $$n$$ and $$k$$ are positive integers. In order to maximize the value of $$m$$, $$x$$ should be of the form:

Note: the exponent of $$x_m$$ must be minimized.

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