Remote-sensing visual question answering (RSVQA) aims to provide accurate answers to remote sensing images and their associated questions by leveraging both visual and textual information during the inference process. However, most existing methods ignore the significance of the interaction between visual and language features, which typically adopt simple feature fusion strategies and fail to adequately model cross-modal attention, struggling to capture the complex semantic relationships between questions and images. In this study, we introduce a unified transformer with cross-modal mixture expert (TCMME) model to address the RSVQA problem. Specifically, we utilize the vision transformer (VIT) and BERT to extract visual and language features, respectively. Furthermore, we incorporate cross-modal mixture experts (CMMEs) to facilitate cross-modal representation learning. By leveraging the shared self-attention and cross-modal attention within CMMEs, as well as the modality experts, we effectively capture the intricate interactions between visual and language features and better focus on their complex semantic relationships. Finally, we conduct qualitative and quantitative experiments on two benchmark datasets: RSVQA-LR and RSVQA-HR. The results demonstrate that our proposed method surpasses the current state-of-the-art (SOTA) techniques. Additionally, we perform an extensive analysis to validate the effectiveness of different components in our framework.