Solve$(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$

Question

We need to solve the following equation $y=(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$ and it is equal to 2 while I am getting the value in excel I am not able to solve it manually eventhough the values are conjugate I tried $y=a-b$

$a=(10+6\sqrt3)^{\frac{1}{3}}$ & $b=(-10+6\sqrt3)^{\frac{1}{3}}$

$y^3=(a-b)^3$

$y^3=a^3-b^3-3ab(a-b)$ after this step I am struck

Answer 1

Substituting the values of $a$ and $b$, and the relation $y=a-b$, into $y^3=a^3-b^3-3ab(a-b)$ yields $$y^3=10+6\sqrt3+10-6\sqrt3-3((6\sqrt3+10)(6\sqrt3-10))^{1/3}y$$ $$y^3=20-3(3\cdot36-100)^{1/3}y$$ $$y^3=20-6y$$ $$y^3+6y-20=0$$ $$(y-2)(y^2+2y+10)=0$$ Since the quadratic factor has no real roots, $y=2$.

Answer 2

Since $(1\pm\sqrt3)^3=10\pm6\sqrt3$,$$\sqrt[3]{10+6\sqrt3}+\sqrt[3]{10-6\sqrt3}=1+\sqrt3+1-\sqrt3=2.$$