(1) $f:\mathbb R\to \mathbb R$ is $C^\infty$.

(2) $f^2:\mathbb R\to \mathbb R$ is $C^\infty$.
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EDIT: The right formulation is: If $f\ge 0$ is $C^\infty$, then one can choose a square root of  $f$ which is twice differentiable, but not better in general. 

(3) $f^2$ and $f^3:\mathbb R\to \mathbb R$ are both $C^\infty$.

(4) $f^p$ and $f^q:\mathbb R\to \mathbb R$ are both $C^\infty$.
EDIT: Where $p$, $q$ are relatively prime. 

Obviously, (1) $\implies$ (3), (4).

(2) 
See:<BR>
Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Choosing roots of polynomials smoothly, Israel J. Math 105 (1998), p. 203-233.[(pdf)][1]

(3) $\implies$ (1). See: <BR> MR0682456 
Joris, Henri:
Une $C^\infty$-application non-immersive qui possède la propriété universelle des immersions. Arch. Math. (Basel) 39 (1982), no. 3, 269–277.
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and:
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MR2179865 Myers, Robert: An elementary proof of Joris's theorem. Amer. Math. Monthly 112 (2005), no. 9, 829–831.  

(4) $\implies$ (1). See:<BR>
MR0833407  Duncan, John; Krantz, Steven G.; Parks, Harold R.: Nonlinear conditions for differentiability of functions. J. Analyse Math. 45 (1985), 46–68.


  [1]: http://www.mat.univie.ac.at/~michor/roots.pdf