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One of our formula for radius of atom was $\frac{n^2h^2}{4\pi^2me^2}$.

And another one was,

$\frac{n^2h^2}{4\pi^2me^2K}$.

For hydrogen,

Why $k_e$ is sometimes included and sometimes excluded ?

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Your first equation is written using Gaussian units while the second is written using SI units. Either is a perfectly valid way to write down the equation provided you make sure to stick to whatever system of units you are using.

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