  In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence
form elliptic fourth-order operator $A_\varepsilon$ with measurable
$\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter.
For the resolvent $(A_\varepsilon+1)^{-1}$, acting as an operator from $L^2$ to
$H^2$, we find an approximation with remainder term of order $O(\varepsilon^2)$
as $\varepsilon$ tends to $0$. Relying on this result, we construct the
resolvent approximation with remainder of order $O(\varepsilon^3)$ in the
operator $L^2$-norm. We employ two-scale expansions that involve smoothing.
