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<abstract><p>An entropy measure of uncertainty has a complementary dual function called extropy. In the last six years, this measure of randomness has gotten a lot of attention. It cannot, however, be applied to systems that have survived for some time. As a result, the idea of residual extropy was created. To estimate the extropy and residual extropy, Bayesian and non-Bayesian estimators of unknown parameters of the exponentiated gamma distribution are generated. Bayesian estimators are regarded using balanced loss functions like the balanced squared error, balanced linear exponential and balanced general entropy. We use the Lindley method to get the extropy and residual extropy estimates for the exponentiated gamma distribution based on generalized type-Ⅰ hybrid censored data. To test the effectiveness of the proposed methodologies, a simulation experiment was carried out, and the actual data set was studied for illustrative purposes. In summary, the mean squared error values decrease as the number of failures increases, according to the results obtained. The Bayesian estimates of residual extropy under the balanced linear exponential loss function perform well compared to the other estimates. Alternatively, the Bayesian estimates of the extropy perform well under a balanced general entropy loss function in the majority of situations.</p></abstract>