Question

question
rewrite \\(\sqrt3{192}\\) in simplest form.

Explanation:

Step1: Factor 192 into prime factors

We need to find the prime factorization of 192. We start by dividing by 2 repeatedly:
\(192\div2 = 96\)
\(96\div2 = 48\)
\(48\div2 = 24\)
\(24\div2 = 12\)
\(12\div2 = 6\)
\(6\div2 = 3\)
So, \(192 = 2^6\times3\)

Step2: Rewrite the cube root using the prime factors

We know that \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\) (for real numbers \(a,b\)). So we can rewrite \(\sqrt[3]{192}\) as \(\sqrt[3]{2^6\times3}\).
Using the property \(\sqrt[3]{a^n}=a^{\frac{n}{3}}\) (when \(n\) is a multiple of 3), we have \(\sqrt[3]{2^6}=2^{\frac{6}{3}} = 2^2=4\)
So, \(\sqrt[3]{2^6\times3}=\sqrt[3]{2^6}\times\sqrt[3]{3}=4\sqrt[3]{3}\)

Answer:

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