<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] 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, $y=2$.
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