Suppose that $0<\alpha_1<\cdots<\alpha_n<1$. Let $M$ be a smooth paracompact connected manifold of dimension $d$. Let $(U_k)_{k\in\mathbb{N}}$ be a locally finite open covering of $M$. Suppose that each $U_k$ is relatively compact.
Is it true that there exists a partition of unity $(\zeta_k)_{k\in\mathbb{N}}$ on $M$ subordinates to the open covering $(U_k)_k$ such that for each $k\in\mathbb{N}$ $\zeta_k$ and $\zeta_k^{\alpha_1}$ and $\cdots$ $\zeta_k^{\alpha_n}$ are smooth?
I want to know especially the result when $\alpha_1=\frac{1}{4}, \alpha_2=\frac{1}{2}$.