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[tex]KE=mc^2(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1)[/tex] (1)

For speeds much lower than the speed of light however, we know that [tex]KE=\frac{1}{2}mv^2[/tex] (2)

The second expression can be derived from the first one using binomial expansion of the term with the square root in the denominator. I can see how the math behind this works, so that is not my question. What I'm wondering is rather WHY (again, I don't mean the mathematical reason but rather a physical reason) doing this operation on the equation for relativistic kinetic energy leads to the equation non-relativistic kinetic energy?

I mean if we look at the first expression, the term with the square root has the exponent -(1/2). This exponent is apparently constant, not variable. So from where/what would one get the idea of binomally expanding it like it was variable? Where does the intuition behind this come from?

Hope my question is understandable.