Given an integer-valued function $f:\{0,1\}^n\rightarrow \{0,1,\dots, m-1\}$ that is mildly hard to compute on instances drawn from some distribution $D$ over $\{0,1\}^n$, we show that the function $g(x_1, \dots, x_t) = f(x_1) + \dots + f(x_t)$ is strongly hard to compute on instances $(x_1, \dots, x_t)$ drawn from the product distribution $D^t$. We also show the same for the task of approximately computing real-valued functions $f: \{0,1\}^n \rightarrow [0,m)$. Our theorems immediately imply hardness self-amplification for several natural problems including Max-Clique and Max-SAT, Approximate $\#$SAT, Entropy Estimation, etc..