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(Sqrt[72]/(Sqrt[26]+Sqrt[2])^12=a+bSqrt[13] a+3b=1889 It’s in my head.
Reach+++++++++
In the statement of the problem it must be given that the numbers a and b are rational. Otherwise the solution is very nice
√(72)/(√(26) + √2)= ( √(13 ) - 1)/2 = x, say x ^2 + x = 3 x^6 + x^3 + 3 x ^3 * ( x^2 + x) = 27x^6 + x^3 + 9 x ^3 = 27x^6 + 10 x^3 = 27 x ^ 12 = 27 - 10 x ( 3 - x) = 27 - 30 x + 10 ( 3 - x) = 57 - 40 x = 57 - 20 ( √ (13) - 1) = 77 - 20 √ (13)
(Sqrt[72]/(Sqrt[26]+Sqrt[2])^12=a+bSqrt[13] a+3b=1889 It’s in my head.
Reach+++++++++
In the statement of the problem it must be given that the numbers a and b are rational. Otherwise the solution is very nice
√(72)/(√(26) + √2)
= ( √(13 ) - 1)/2 = x, say
x ^2 + x = 3
x^6 + x^3 + 3 x ^3 * ( x^2 + x) = 27
x^6 + x^3 + 9 x ^3 = 27
x^6 + 10 x^3 = 27
x ^ 12 = 27 - 10 x ( 3 - x)
= 27 - 30 x + 10 ( 3 - x)
= 57 - 40 x
= 57 - 20 ( √ (13) - 1)
= 77 - 20 √ (13)