what is $sqrt3{-512}$?

Explanation:

Step1: Recall cube - root definition

We need to find a number $x$ such that $x^3=-512$.

Step2: Factorize 512

$512 = 2\times2\times2\times2\times2\times2\times2\times2\times2=2^9$.

Step3: Consider the negative sign

Since $(-8)\times(-8)\times(-8)=-(8\times8\times8)= - 512$ and $8 = 2^3$, then $\sqrt[3]{-512}=-8$.

Answer:

$-8$