How do you want to simplify this? If its in terms of common factors multiplication wise then I don't think you can do that.
Question directions simply state to simplify: (hehe, simply simplify)
(20+14sqrt(2))^(1/3) + (20-14sqrt(2))^(1/3)
WolframAlpha solution says to express 20 + 14√(2) as a cube to equal 8 + 12√(2) + 6((√2))^2+(√(2))^3. Can someone explain how these numbers were crunched up? Is it to do with (a^3+b^3)=(a+b)(a^2-ab+b^2)?
As far as i know this kind of thing relies a lot on luck. We can try
cube both sides to get
Now we see
Luckily , we find with very little effort, a = 2 , b = 1 works so
Repeating the procedure using the right hand side of $(1) , you don't have to start from scratch , we find the other cube root must satisfy
And again with little effort we find a = 2 , b = -1 works so
So the sum of your cube roots is actually the very humble integer 4.