  We study the cosmological and weak-field properties of theories of gravity
derived by extending general relativity by means of a Lagrangian proportional
to $R^{1+\delta}$. This scale-free extension reduces to general relativity when
$\delta \to 0$. In order to constrain generalisations of general relativity of
this power class we analyse the behaviour of the perfect-fluid Friedmann
universes and isolate the physically relevant models of zero curvature. A
stable matter-dominated period of evolution requires $\delta >0$ or $\delta
<-1/4$. The stable attractors of the evolution are found. By considering the
synthesis of light elements (helium-4, deuterium and lithium-7) we obtain the
bound $-0.017<\delta <0.0012.$ We evaluate the effect on the power spectrum of
clustering via the shift in the epoch of matter-radiation equality. The horizon
size at matter--radiation equality will be shifted by $\sim 1%$ for a value of
$\delta \sim 0.0005.$ We study the stable extensions of the Schwarzschild
solution in these theories and calculate the timelike and null geodesics. No
significant bounds arise from null geodesic effects but the perihelion
precession observations lead to the strong bound $\delta =2.7\pm 4.5\times
10^{-19}$ assuming that Mercury follows a timelike geodesic. The combination of
these observational constraints leads to the overall bound $0\leq \delta
<7.2\times 10^{-19}$ on theories of this type.
