Question

what is $sqrt3{512}$?

Explanation:

Step1: Recall cube - root definition

The cube - root of a number \(x\), denoted as \(\sqrt[3]{x}\), is a number \(y\) such that \(y^3=x\). We need to find a number \(y\) where \(y\times y\times y = 512\).

Step2: Factorize 512

We know that \(512=2\times256=2\times2\times128=2\times2\times2\times64=2\times2\times2\times2\times32=2\times2\times2\times2\times2\times16=2\times2\times2\times2\times2\times2\times8=2\times2\times2\times2\times2\times2\times2\times4=2\times2\times2\times2\times2\times2\times2\times2\times2 = 2^9\).

Step3: Calculate the cube - root

\(\sqrt[3]{512}=\sqrt[3]{2^9}\). According to the rule \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), when \(n = 3\) and \(m = 9\), we have \(\sqrt[3]{2^9}=2^{\frac{9}{3}}=2^3\). And \(2^3=2\times2\times2 = 8\).

Answer:

8