Show that α=³√(7+5√2)+³√(7-5√2) is an integer. No calculators or computers allowed!

a = cuberoot(7+ 5V2) + cuberoot(7 - 5v2)

or, a^3 = 14 + 3*cuberoot(49-50)*a

or, a^3 + 3a - 14 = 0

or, (a-2)(a^2 +5a+7) = 0

or, a =2, disregarding the complexroots of a which are inadmissible.

Consequently, it follows that a is an integer.

Posted by K Sengupta on 2007-05-16 11:16:02