A method for synthesizing the desired sound field while suppressing the exterior radiation power with directional weighting is proposed. The exterior radiation from the loudspeakers in sound field synthesis systems can be problematic in practical situations. Although several methods to suppress the exterior radiation have been proposed, suppression in all outward directions is generally difficult, especially when the number of loudspeakers is not sufficiently large. We propose the directionally weighted exterior radiation representation to prioritize the suppression directions by incorporating it into the optimization problem of sound field synthesis. By using the proposed representation, the exterior radiation in the prioritized directions can be significantly reduced while maintaining high interior synthesis accuracy, owing to the relaxed constraint on the exterior radiation. Its performance is evaluated with the application of the proposed representation to amplitude matching in numerical experiments.