Finding roots of a mathematical equation If $x^3 + 3^x = 0$ what will be the value of $x$?
what will be the easiest way to find it mathematically? 

$$x^3 + 3^x = 0 \iff x^3 = -3^x$$
If $x$ is real, $$\implies x = -3^{\frac{x}{3}} \implies3 ^{-\frac{x}{3}} x = -1 \implies xe^{-\frac{x}{3} \ln3} = -1 \implies -\frac{x}{3} \ln(3) e^{-\frac{x}{3} \ln3} = \frac{\ln 3}{3}$$
So $$-\frac{x}{3} \ln 3 = W(\frac{\ln 3}{3})$$
$$x = - \frac{3W(\frac{\ln 3}{3})}{\ln 3 }$$
If x is complex, $$x = -3^{\frac{x}{3}} \omega $$ or $$x = -3^{\frac{x}{3}} \omega^2 $$ where $\omega$ is a primitive root of order $3$.
In the same way, one will get $$x = - \frac{3W(-\frac{1}{6} i (-i \ln 3 + \sqrt 3 \ln 3 ))}{\ln 3 }$$ and $$x = - \frac{3W(\frac{1}{6} i (i \ln 3 + \sqrt 3 \ln 3 ))}{\ln 3 }$$
Where W is Lambert W function and $i$ is unity of imaginery.
