evaluate the expression.\n$sqrt3{512}$

Explanation:

Step1: Recall cube - root concept

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

Step2: Factorize 512

$512 = 2\times256=2\times2\times128 = 2\times2\times2\times64=2\times2\times2\times2\times32=2\times2\times2\times2\times2\times16=2\times2\times2\times2\times2\times2\times8=2\times2\times2\times2\times2\times2\times2\times4=2\times2\times2\times2\times2\times2\times2\times2\times2 = 2^9$.

Step3: Calculate the cube - root

$\sqrt[3]{512}=\sqrt[3]{2^9}=2^{\frac{9}{3}} = 2^3$. Since $2^3=8$, $\sqrt[3]{512}=8$.

Answer:

8