Human grasps can be roughly categorized into two types: power grasps and precision grasps. Precision grasping enables tool use and is believed to have influenced human evolution. Today's multi-fingered robotic hands are effective in power grasps, but for tasks requiring precision, parallel grippers are still more widely adopted. This contrast highlights a key limitation in current robotic hand design: the difficulty of achieving both stable power grasps and precise, fine-grained manipulation within a single, versatile system. In this work, we bridge this gap by jointly optimizing the control and hardware design of a multi-fingered dexterous hand, enabling both power and precision manipulation. Rather than redesigning the entire hand, we introduce a lightweight fingertip geometry modification, represent it as a contact plane, and jointly optimize its parameters along with the corresponding control. Our control strategy dynamically switches between power and precision manipulation and simplifies precision control into parallel thumb-index motions, which proves robust for sim-to-real transfer. On the design side, we leverage large-scale simulation to optimize the fingertip geometry using a differentiable neural-physics surrogate model. We validate our approach through extensive experiments in both sim-to-real and real-to-real settings. Our method achieves an 82.5% zero-shot success rate on unseen objects in sim-to-real precision grasping, and a 93.3% success rate in challenging real-world tasks involving bread pinching. These results demonstrate that our co-design framework can significantly enhance the fine-grained manipulation ability of multi-fingered hands without reducing their ability for power grasps. Our project page is at https://jianglongye.com/power-to-precision