We experimentally determine the radiative quantum efficiency of bright excitons in carbon nanotubes by modifying the exciton dynamics through cavity quantum electrodynamical effects [1]. Specially designed air-mode silicon nanobeam cavities [2,3] are used to obtain efficient coupling to individual air-suspended carbon nanotubes. Spectral and temporal behavior of the cavity enhancement is characterized by photoluminescence spectroscopy at room temperature, and the fraction of the radiative decay process is evaluated. We find that the radiative quantum efficiency is near unity for the bright excitons in suspended nanotubes.Work supported in part by MIC (SCOPE 191503001), JSPS (KAKENHI JP20H02558, JP20J00817, JP20K15199), MEXT (Nanotechnology Platform JPMX09F19UT0072), and RIKEN (Incentive Research Project). H.M. acknowledges support from RIKEN (Junior Research Associate Program). Finite-difference time-domain calculations are performed using HOKUSAI BigWaterfall supercomputer at RIKEN. We thank the Advanced Manufacturing Support Team at RIKEN for technical assistance.[1] H. Machiya, D. Yamashita, A. Ishii, Y. K. Kato, in preparation.[2] R. Miura, S. Imamura, R. Ohta, A. Ishii, X. Liu, T. Shimada, S. Iwamoto, Y. Arakawa, Y. K. Kato, Nat. Commun. 5, 5580 (2014).[3] H. Machiya, T. Uda, A. Ishii, Y. K. Kato, Appl. Phys. Lett. 112, 021101 (2018).