How can I compute the chern roots of a vector bundle using the characteristic classes?

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Let $E \to X$ be an algebraic vector bundle over a complex projective variety $X$. If I know the Chern classes of $E$ is there a way to find the Chern roots from them? If so, how can I do this? For example, what are the chern roots of $\Omega_X$ for $X$ a K3 surface?