Simplifying a simple rational function

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For this expression, you can use simplify_rational() instead of simplify():

sage: var('d, T')
(d, T)
sage: f = (d^(T+1)-d)/d
sage: f.simplify_rational()
d^T - 1

Note that in this case you also have:

sage: f.expand()
d^T - 1

Note that in a more general case, the "most simplified" expression is obtained with the command simplify_full():

sage: f.simplify_full()
d^T - 1