Question

simplify
$2\sqrt3{192}$
\bigcirc $16\sqrt3{3}$
\bigcirc $12\sqrt3{3}$
\bigcirc $8\sqrt3{3}$
\bigcirc $4\sqrt3{3}$

Explanation:

Step1: Factor 192 for cube root

We know that to simplify a cube root, we factor the number inside the cube root into perfect cubes and other factors. Let's factor 192. We start by dividing by the smallest perfect cube factor. The perfect cube factors are \(1, 8, 27, 64, 125,\cdots\). We see that \(192 = 64\times3\), and \(64 = 4^3\) (since \(4\times4\times4=64\)). So we can write \(\sqrt[3]{192}=\sqrt[3]{64\times3}\).

Using the property of cube roots \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\) (where \(a = 64\) and \(b = 3\)), we get \(\sqrt[3]{64\times3}=\sqrt[3]{64}\times\sqrt[3]{3}\). Since \(\sqrt[3]{64} = 4\) (because \(4^3=64\)), then \(\sqrt[3]{192}=4\sqrt[3]{3}\).

Step2: Multiply by the coefficient outside

Now we have the original expression \(2\sqrt[3]{192}\). Substitute the simplified cube root we found in step 1: \(2\times(4\sqrt[3]{3})\).

When we multiply the coefficients \(2\) and \(4\), we get \(2\times4 = 8\). So the expression becomes \(8\sqrt[3]{3}\).

Answer:

\(8\sqrt[3]{3}\) (corresponding to the option with \(8\sqrt[3]{3}\))