Cloud services have been widely employed in IT industry and scientific research. By using Cloud services users can move computing tasks and data away from local computers to remote datacenters. By accessing Internet-based services over lightweight and mobile devices, users deploy diversified Cloud applications on powerful machines. The key drivers towards this paradigm for the scientific computing field include the substantial computing capacity, on-demand provisioning and cross-platform interoperability. To fully harness the Cloud services for scientific computing, however, we need to design an application-specific platform to help the users efficiently migrate their applications. In this, we propose a Cloud service platform for symbolic-numeric computation - SNC. SNC allows the Cloud users to describe tasks as symbolic expressions through C/C++, Python, Java APIs and SNC script. Just-In-Time (JIT) compilation through using LLVM/JVM is used to compile the user code to the machine code. We implemented the SNC design and tested a wide range of symbolic-numeric computation applications (including nonlinear minimization, Monte Carlo integration, finite element assembly and multibody dynamics) on several popular cloud platforms (including the Google Compute Engine, Amazon EC2, Microsoft Azure, Rackspace, HP Helion and VMWare vCloud). These results demonstrate that our approach can work across multiple cloud platforms, support different languages and significantly improve the performance of symbolic-numeric computation using cloud platforms. This offered a way to stimulate the need for using the cloud computing for the symbolic-numeric computation in the field of scientific research.