Question

(\frac{12(sqrt3{54})}{3(sqrt3{6})})

Explanation:

Step1: Simplify the cube root of 54

First, factorize 54. We know that \(54 = 27\times2=3^{3}\times2\). So, \(\sqrt[3]{54}=\sqrt[3]{3^{3}\times2}=3\sqrt[3]{2}\) (using the property \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\) for \(a = 3^{3}\) and \(b = 2\)).

Step2: Simplify the cube root of 6

The cube root of 6, \(\sqrt[3]{6}\) remains as it is for now.

Step3: Simplify the fraction of the coefficients and the cube roots

The given expression is \(\frac{12\sqrt[3]{54}}{3\sqrt[3]{6}}\). Substitute \(\sqrt[3]{54}=3\sqrt[3]{2}\) into the numerator:
\[

$$\begin{align*} \frac{12\times3\sqrt[3]{2}}{3\sqrt[3]{6}}&=\frac{36\sqrt[3]{2}}{3\sqrt[3]{6}}\\ &=\frac{12\sqrt[3]{2}}{\sqrt[3]{6}} \end{align*}$$

\]
Rationalize the denominator by multiplying the numerator and denominator by \(\sqrt[3]{6^{2}}\) (since to rationalize a cube root \(\sqrt[3]{a}\), we multiply by \(\sqrt[3]{a^{2}}\) to get \(a\) in the denominator).
\[

$$\begin{align*} \frac{12\sqrt[3]{2}\times\sqrt[3]{6^{2}}}{\sqrt[3]{6}\times\sqrt[3]{6^{2}}}&=\frac{12\sqrt[3]{2\times6^{2}}}{\sqrt[3]{6^{3}}}\\ &=\frac{12\sqrt[3]{2\times36}}{6}\\ &=\frac{12\sqrt[3]{72}}{6}\\ & = 2\sqrt[3]{72} \end{align*}$$

\]
Wait, maybe a simpler way. Let's go back. We know that \(\sqrt[3]{54}=\sqrt[3]{27\times2} = 3\sqrt[3]{2}\) and \(\sqrt[3]{6}\) is just \(\sqrt[3]{6}\). So the original expression:
\[
\frac{12\times3\sqrt[3]{2}}{3\times\sqrt[3]{6}}=\frac{12\sqrt[3]{2}}{\sqrt[3]{6}}=\frac{12\sqrt[3]{\frac{2}{6}}}{1}=\frac{12\sqrt[3]{\frac{1}{3}}}{1}
\]
Wait, no, \(\frac{\sqrt[3]{2}}{\sqrt[3]{6}}=\sqrt[3]{\frac{2}{6}}=\sqrt[3]{\frac{1}{3}}\). But maybe we made a mistake in the first step. Let's re - express the original fraction:

The original expression is \(\frac{12\sqrt[3]{54}}{3\sqrt[3]{6}}\). First, simplify the coefficient \(\frac{12}{3}=4\). Then we have \(4\times\frac{\sqrt[3]{54}}{\sqrt[3]{6}}\). Since \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}\), so \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\sqrt[3]{\frac{54}{6}}=\sqrt[3]{9}\). So the expression becomes \(4\sqrt[3]{9}\).

Ah, that's a simpler way. Because \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\sqrt[3]{\frac{54}{6}}=\sqrt[3]{9}\), and \(\frac{12}{3} = 4\). So \(4\times\sqrt[3]{9}=4\sqrt[3]{9}\).

Let's check:

\(\sqrt[3]{54}\div\sqrt[3]{6}=\sqrt[3]{\frac{54}{6}}=\sqrt[3]{9}\), and \(12\div3 = 4\). So the simplified form is \(4\sqrt[3]{9}\).

Answer:

\(4\sqrt[3]{9}\)