a = 2 + √3 1/a = 1/(2 + √3) = 2 - √3 a + 1/a = 2 + √3 + 2 - √3 a + 1/a = 4 (a^6+a^4+a^2+1)/a^3 → (a^3 (a^3+a+1/a+1/a^3 ))/a^3 = a^3+a+1/a+1/a^3 = (a^3+1/a^3 ) + (a+1/a) a+1/a = 4 On cubing both sides, (a+1/a)^3 = (4)^3 a^3+1/a^3 + 3 × a×1/a (4) = 64 a^3+1/a^3 + 12 = 64 a^3+1/a^3 = 52 Therefore, a^3+1/a^3 + a+1/a = 52 + 4 = 56.
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