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One possible trick is to replace sqrt(2)
by a variable t
and sqrt(3)
by a variable d
, so you get a polynomial in t
and d
, and introduce a variable z
for sqrt(6)
, and then reduce the polynomial in t
and d
by t*d - z
using some monomial ordering where t*d > z
, e.g. the lexicographic ordering with t > d > z
, so that all instances of t*d
are replaced by z
, and then you can substitute z
by sqrt(6)
:
def collect_sqrt2sqrt3_to_sqrt6(expr):
expr_poly = expr.subs({sqrt(2) : var('t'), sqrt(3) : var('d')})
R = PolynomialRing(SR, names=['t', 'd', 'z'], order='lex')
t, d, z = R.gens()
poly = expr_poly.polynomial(ring=R)
poly_collected = poly.reduce([t*d - z])
return SR(poly_collected.subs({z : sqrt(6), t : sqrt(2), d : sqrt(3)}))
Example:
sage: a = sqrt(2)*sqrt(3)+6
sage: collect_sqrt2sqrt3_to_sqrt6(a)
sqrt(6) + 6
With some more effort this could be generalized to detect which square roots appear in the expression, etc.