Let x = (5-2sqrt(13))^(1/3) the quantity in question. You can make the ansatz 5-2sqrt(13) =x^3:= (a - b sqrt(13))^3 which gives for a and b the two equations (1) 5=a^3+39 a b^2 and (2) 2 = 3 a^2 b +13 b^3. We could try b = a and find from (1) and (2): (1') 5 = a^3(1+39) = 40 a^3 and (2') 2 = a^3(3+13) = 16 a^3 from which we find a=b=1/2. Hence x =( (1-sqrt(13))/2).
Let x = (5-2sqrt(13))^(1/3) the quantity in question.
You can make the ansatz 5-2sqrt(13) =x^3:= (a - b sqrt(13))^3 which gives for a and b the two equations
(1) 5=a^3+39 a b^2 and
(2) 2 = 3 a^2 b +13 b^3.
We could try b = a and find from (1) and (2):
(1') 5 = a^3(1+39) = 40 a^3 and
(2') 2 = a^3(3+13) = 16 a^3
from which we find
a=b=1/2.
Hence x =( (1-sqrt(13))/2).
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