Question

∛{7+5√2} + ∛{7-5√2} =?

Answer 1

let  ∛(7 +5√2)    =  a  =  x ( 1 + y√2) 
 and  ∛(7 - 5√2)  = b

So  7 + 5√2  = x³ (1 + y √2)³ 
   =  x³ (1 + 6 y²) + √2 * x³ (3 y+ 2 y³)
equate rational and irrational parts.
    (1 + 6 y²) = 7 / x³    --- (1)
    (3 y + 2 y³) = 5 / x³    ---- (2)

Solving these two equations we get: 
   14 y³ - 30 y² + 21 y - 5 = 0
   looking at the coefficients :  14+21-30-5=0

 so  y = 1 is a solution.  then  we get:  14 y² - 16 y + 5 as the other factor.    The roots of this polynomial are imaginary.

Thus we get y = 1.  hence by (1),    1 + 6 *1² =7/x³     =>  x = 1

so  ∛(7 + 5√2) =  1 + √2

similarly  ∛(7 - 5√2) = 1 - √2

the answer is  2.