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You can try this:

sage: x, a = var('x a') 
sage: expr = x^a/x      
sage: expr.full_simplify()
x^(a - 1)

You Concerning your first question, you can try this:

sage: x, a = var('x a') 
sage: expr = x^a/x      
sage: expr.full_simplify()
x^(a - 1)