Simplifying a Golden Radical Expression in Three Ways

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[ahmadmazbouh]

you always leave the beauty and the greatness to the last method in the video

[Grassmpl]

Comment regarding your 3rd method: you know that a, b nust be multiples of 1/2 which differ by an integer. This is because cbrt(2+sqrt 5) is satisfied by a monic integer polynomial. Thus, it is an algebraic integer. However, the number field Q(sqrt5) has an integral basis {1, (1+ sqrt 5)/2}.

[MichaelRothwell]

Nice video! As I was only able to come up with 2 methods to solve this problem, I was looking forward to learn a third method. My disappointment in this regard was well compensated by seeing the b=ka substitution in the equations with homogeneous LHS, leading to a simple cubic (I found the cube root by an inspired guess at the answer and checking, a less robust method). I do think though that in method 3 it would have been prudent to check that the result found is in fact a cube root of 2+√5.

[imadbarakathaji]

Amazing 🤩
What is name of the board that you use??

[XJWill]

cbrt(2 ± sqrt(5)) = 1/2 ± sqrt(5)/2 so the answer is 1

[tonyhaddad]

Great problem, all method are awesome good job

[francis]

I just rearranged the expressions in the cube roots to get cbrt(phi^3) + cbrt((-1/phi)^3)

[itsphoenixingtime]

This is actually how Cardano's formula expression looks like :)

[JohnRandomness]

My method: this is obviously a cubic equation's solution. So go back to the cubic equation by cubing ?. We get ?³ + 3? = 4. The only real solution is ? = 1.

It's quite ironic that to solve a cubic equation, we need to take the solution, return to the cubic equation, and resolve it. Or possibly go to a different cubic equation.

[maismullimindrslri]

здравствуйте Я учился в Украине.Вы очень сложно решаете.Приравняем иксу умножаем обе стороны на два внеснм двойку под корень.шестьнадцать плюсь восемь корень пять.это польный куб.

[giuseppemalaguti]

Pongo la somma =t, elevo al cubo, e lavorando un po', ha una cubica come equazione finale. t^3+3t-4=0 che mi dà come unica soluzione reale t=1

[arekkrolak]

cube root of negative 1 has three solutions, like every root of n-th degree has n solutions :)

[markobavdek]

first equation... f(x)=0 = sad face ....I did not like it

[rishiitjecrc]

Let cubrt(2 + sqrt(5)) + cubrt(2 - sqrt(5)) =t
Cube both side
Cube(t) = 2+sqrt(5) + 2 -sqrt(5) + 3(cubrt(4-5) x (cubrt(2 + sqrt(5)) + cubrt(2 - sqrt(5)))
Cube(t) = 2 + 2 + 3(-1)(t)
Cube(t) = 4-3t
Cube(t) +3t -4 =0
(t-1)(square (t) +t+4) = 0
t=1
Quadratic has complex soln

[yoav]

This video is number 1! (And 1 and 1😃)

[MrLidless]

Each cube root has three solutions once you include complex numbers, so there are actually nine solutions, not one.