Question

simplify the radical \\(\sqrt3{192}\\):

Explanation:

Step1: Factor 192 into prime factors

First, we factorize 192. We know that \(192 = 64\times3\), and \(64 = 4^3=2^6\). So, \(192 = 2^6\times3\).

Step2: Simplify the cube root

Using the property of radicals \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\) (where \(a = 2^6\) and \(b = 3\)), we have \(\sqrt[3]{192}=\sqrt[3]{2^6\times3}\).
Then, using the property \(\sqrt[3]{a^n}=a^{\frac{n}{3}}\) for \(a = 2\) and \(n = 6\), we get \(\sqrt[3]{2^6}=2^{\frac{6}{3}} = 2^2=4\).
So, \(\sqrt[3]{192}=4\sqrt[3]{3}\).

Answer:

\(4\sqrt[3]{3}\)