how do I solve this?

[hoary lion]

I dont really understand can write here maybe?

[stray cave]

cube both sides

[hallow ingot]

n = p/2
8 = 216/27
3root(p/2 + sqrt(p^2/4 + 216/27)) + 3root(p/2 - sqrt(p^2/4 + 216/27)) = 8
This is Cardano’s formula, a formula to solve cubics in the form x^3 + bx + q
||
In this case, q is p, or 2n, and b is 6
x^3 + 6x + 2n = 0
X = 8
512 + 48 + 2n = 0
n = -280||

[ruby dew]

Let $a = n+\sqrt{n^2+8}$ and $b = n-\sqrt{n^2+8}$. It is given that $\sqrt[3] a+\sqrt[3] b = 8$.\medskip \
Note that $ab = n^2-(n^2+8) = -8$. \medskip \
Since $(\sqrt[3] a+\sqrt[3] b)^3 = a+b+3\sqrt[3]{ab}(\sqrt[3] a + \sqrt[3]{b})$, we have $$8 = a+b+3\cdot \sqrt[3]{-8} \cdot 8. $$ So, $8 = 2n +3\cdot (-2) \cdot 8$. Hence, $2n = 56$ and $n=28$