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Computations in algebraic numbers are costly so
Sage computes the product of a and b "lazily".
Some computations force it to realise the exact value of c.
Here are the one I would favour.
sage: c = AA(r) * AA(1/r)
sage: c
1.000000000000000?
sage: c.exactify()
sage: c
1
For the sake of completeness I also include those from @rburing's answer:
sage: c = AA(r) * AA(1/r)
sage: c
1.000000000000000?
sage: c.radical_expression()
1
sage: c = AA(r) * AA(1/r)
sage: c
1.000000000000000?
sage: c.minpoly()
x - 1
sage: c
1
