This paper investigates the geometry of the expansion ${}_Q$ of the real field ℝ by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński-Lion-Rolin). To this end, we study non-standard models of the universal diagram T of ${}_Q$ in the language ℒ augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation...