Question

Prove that the real root of the equation $x^3+12x-12=0$ is $2\sqrt[3]{32}-\sqrt[8]{84}$.

Answer 1

Given equation, $x^2+12x-12=0$ and given root is $2\sqrt[3]{32}-\sqrt[3]{34}$

Simplifying the given roots, we can write $root\left(r\right)=2^{\frac{4}{3}}–2^{\frac{2}{3}}$

Substituting $x=r$ in the given equation, we can solve it as:

$(2^{\frac{4}{3}}–2^{\frac{2}{3}}{)}^3+12(2^{\frac{4}{3}}–2^{\frac{2}{3}})-12$

$⟹2^4-3\cdot 2^3\cdot 2^{\frac{1}{3}}+3\cdot 2^2\cdot 2^{\frac{2}{3}}-2^2+3\cdot 2^3\cdot 2^{\frac{1}{3}}-3\cdot 2^2\cdot 2^{\frac{2}{3}}-12=16-4-12=0$

Since the equation reduces to zero for $x=r$, we can say that r is the real root of the given equation.