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Share-Based Compensation (Narrative) (Details) - USD ($)
$ in Millions
3 Months Ended 9 Months Ended
May 31, 2022
May 31, 2021
May 31, 2022
May 31, 2021
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]        
Share-based compensation expense $ 1.7 $ 1.0 $ 4.3 $ 5.3