<s>[INST] Solve$(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$ 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
 [/INST] Since $(1\pm\sqrt3)^3=10\pm6\sqrt3$,$$\sqrt[3]{10+6\sqrt3}+\sqrt[3]{10-6\sqrt3}=1+\sqrt3+1-\sqrt3=2.$$
</s>