Why $\sqrt[3]{{2 + \sqrt 5 }} + \sqrt[3]{{2 - \sqrt 5 }}$ is a rational number? [duplicate]
Why $\sqrt[3]{{2 + \sqrt 5 }} + \sqrt[3]{{2 - \sqrt 5 }}$ is a rational number?
Solution 1:
Let $$\sqrt[3]{{2 + \sqrt 5 }} + \sqrt[3]{{2 - \sqrt 5 }}=x$$ $$(a+b)^3=a^3+b^3+3ab(a+b)$$ Then $$2 + \sqrt 5+2 - \sqrt 5-3x=x^3$$ $$x^3+3x=4$$ $$x=1$$ $$\sqrt[3]{{2 + \sqrt 5 }} + \sqrt[3]{{2 - \sqrt 5 }}=1$$