We prove in this article that almost all large integers have a representation as the sum of a cube, a biquadrate, ..., and a tenth power.

New bounds are given for the exponential sum$$\sum _{P\le p\<2P}^{}e\left(\alpha p^k\right)$$were $k\ge 5,p$ denotes a prime and $e\left(x\right)=exp\left(2\pi ix\right)$.