Question

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

Explanation:

Step1: Simplify the fraction of coefficients

First, simplify the fraction of the coefficients \(\frac{12}{3}\).
\(\frac{12}{3} = 4\)

Step2: Simplify the cube - root terms

We know that \(54=6\times9 = 6\times3^{2}\), and we want to simplify \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}\). Using the property of cube - roots \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}\) (\(b eq0\)), we have:
\(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\sqrt[3]{\frac{54}{6}}=\sqrt[3]{9}\)

Step3: Combine the results

Now, multiply the result of the coefficient simplification and the cube - root simplification. The original expression \(\frac{12(\sqrt[3]{54})}{3(\sqrt[3]{6})}\) is equal to \(4\times\sqrt[3]{9}\), and \(\sqrt[3]{9}=\sqrt[3]{3^{2}}\), but we can also note that \(9 = 3\times3\), and we can rewrite \(4\sqrt[3]{9}\) as \(4\sqrt[3]{3\times3}\), or we can rationalize or simplify further. Wait, actually, when we simplify \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}\), since \(54 = 6\times9=6\times3^{2}\), \(\sqrt[3]{54}=\sqrt[3]{6\times3^{2}}=\sqrt[3]{6}\times\sqrt[3]{3^{2}}\), so \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\sqrt[3]{3^{2}}=\sqrt[3]{9}\) is correct. But let's check again:
\(\frac{12}{3}=4\), and \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\sqrt[3]{\frac{54}{6}}=\sqrt[3]{9}\), so the expression simplifies to \(4\sqrt[3]{9}\). But we can also write \(9 = 3\times3\), and if we want to express it in terms of \(\sqrt[3]{3}\), we know that \(\sqrt[3]{9}=\sqrt[3]{3^{2}}\), so \(4\sqrt[3]{9}=4\times3^{\frac{2}{3}}\). Alternatively, we can note that \(54 = 27\times2=3^{3}\times2\)? Wait, no, \(3^{3}=27\), \(27\times2 = 54\), oh! I made a mistake earlier. \(54=27\times2 = 3^{3}\times2\), so \(\sqrt[3]{54}=\sqrt[3]{3^{3}\times2}=3\sqrt[3]{2}\), and \(\sqrt[3]{6}=\sqrt[3]{2\times3}\). Then \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\frac{3\sqrt[3]{2}}{\sqrt[3]{2}\times\sqrt[3]{3}}=\frac{3}{\sqrt[3]{3}}\). Multiply numerator and denominator by \(\sqrt[3]{9}\) (since \((\sqrt[3]{3})^{2}=\sqrt[3]{9}\)) to rationalize the denominator: \(\frac{3\times\sqrt[3]{9}}{\sqrt[3]{3}\times\sqrt[3]{9}}=\frac{3\sqrt[3]{9}}{\sqrt[3]{27}}=\frac{3\sqrt[3]{9}}{3}=\sqrt[3]{9}\). Wait, but if \(54 = 3^{3}\times2\), then \(\sqrt[3]{54}=3\sqrt[3]{2}\), and \(\sqrt[3]{6}=\sqrt[3]{2\times3}\), so \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\frac{3\sqrt[3]{2}}{\sqrt[3]{2}\times\sqrt[3]{3}}=\frac{3}{\sqrt[3]{3}}\). Then \(\frac{12}{3}=4\), so the original expression is \(4\times\frac{3}{\sqrt[3]{3}}=\frac{12}{\sqrt[3]{3}}\). Rationalizing the denominator: multiply numerator and denominator by \(\sqrt[3]{9}\), we get \(\frac{12\sqrt[3]{9}}{\sqrt[3]{3}\times\sqrt[3]{9}}=\frac{12\sqrt[3]{9}}{\sqrt[3]{27}}=\frac{12\sqrt[3]{9}}{3} = 4\sqrt[3]{9}\). But another way: since \(54=6\times9 = 6\times3^{2}\), and we can also see that when we simplify \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}\), we can factor \(54\) as \(6\times9\), so \(\sqrt[3]{54}=\sqrt[3]{6}\times\sqrt[3]{9}\), so \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\sqrt[3]{9}\), and \(\frac{12}{3}=4\), so the simplified form is \(4\sqrt[3]{9}\). But we can also write \(4\sqrt[3]{9}\) as \(4\sqrt[3]{3^{2}}\), or if we want to express it with a single cube - root of a perfect cube times something, we know that \(9 = 3\times3\), and there is no perfect cube factor in \(9\) other than \(1\). Wait, but I think I made a mistake in the initial factorization of \(54\). Let's do it correctly: \(54\div6 = 9\), \(6 = 2\times3\), \(9=3\times3\), so \(54 = 2\times3\times3\times3=2\times3^{3}\)? Wait, no, \(3^{3}=27\), \(27\times2 = 54\), yes! \(54 = 3^{3}\times2\), so \(\sqrt[3]{54}=\sqrt[3]{3^{3}\times2}=3\sqrt[3]{2}\), and \(\sqrt[3]{6}=\sqrt[3]{2\times3}\). Then \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\frac{3\sqrt[3]{2}}{\sqrt[3]{2}\times\sqrt[3]{3}}=\frac{3}{\sqrt[3]{3}}\). Then \(\frac{12}{3}=4\), so the expression is \(4\times\frac{3}{\sqrt[3]{3}}=\frac{12}{\sqrt[3]{3}}\). Multiply numerator and denominator by \(\sqrt[3]{9}\): \(\frac{12\sqrt[3]{9}}{\sqrt[3]{3}\times\sqrt[3]{9}}=\frac{12\sqrt[3]{9}}{\sqrt[3]{27}}=\frac{12\sqrt[3]{9}}{3}=4\sqrt[3]{9}\). Now, \(4\sqrt[3]{9}\) can be written as \(4\times9^{\frac{1}{3}}\), or we can note that \(9 = 3^{2}\), so \(4\times(3^{2})^{\frac{1}{3}}=4\times3^{\frac{2}{3}}\). But if we want to simplify it to a form with a single cube - root, we can also say that \(4\sqrt[3]{9}\) is the simplified form, but let's check the calculation again.

Wait, let's start over:

Given the expression \(\frac{12\sqrt[3]{54}}{3\sqrt[3]{6}}\)

Step 1: Simplify the coefficient part \(\frac{12}{3}=4\)

Step 2: Simplify the radical part \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}\). Using the property of radicals \(\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\) (\(n = 3\), \(a = 54\), \(b = 6\))

\(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\sqrt[3]{\frac{54}{6}}=\sqrt[3]{9}\)

Step 3: Multiply the results from step 1 and step 2: \(4\times\sqrt[3]{9}=4\sqrt[3]{9}\)

But we can also simplify \(\sqrt[3]{9}\) as \(\sqrt[3]{3^{2}}\), and if we want to rationalize the denominator (but there is no denominator here now). Alternatively, we can write \(4\sqrt[3]{9}\) as \(4\sqrt[3]{3\times3}\), but the simplest form is \(4\sqrt[3]{9}\) or we can rewrite \(9\) as \(3\times3\) and then we can see that \(4\sqrt[3]{9}=4\sqrt[3]{3^{2}}\). But another way: since \(54 = 27\times2=3^{3}\times2\), \(\sqrt[3]{54}=3\sqrt[3]{2}\), and \(\sqrt[3]{6}=\sqrt[3]{2\times3}\), so \(\frac{\sqrt[3]{54}}{\sqrt[3]{6}}=\frac{3\sqrt[3]{2}}{\sqrt[3]{2}\times\sqrt[3]{3}}=\frac{3}{\sqrt[3]{3}}\). Then multiplying by \(4\) (from \(\frac{12}{3}\)) gives \(\frac{12}{\sqrt[3]{3}}\). Now, rationalize the denominator: multiply numerator and denominator by \(\sqrt[3]{9}\) (because \((\sqrt[3]{3})^{2}=\sqrt[3]{9}\) and \(\sqrt[3]{3}\times\sqrt[3]{9}=\sqrt[3]{27} = 3\))

\(\frac{12\times\sqrt[3]{9}}{\sqrt[3]{3}\times\sqrt[3]{9}}=\frac{12\sqrt[3]{9}}{3}=4\sqrt[3]{9}\)

So the simplified form of the expression \(\frac{12\sqrt[3]{54}}{3\sqrt[3]{6}}\) is \(4\sqrt[3]{9}\) or we can write it as \(4\times3^{\frac{2}{3}}\) (since \(\sqrt[3]{9}=3^{\frac{2}{3}}\))

Answer:

\(4\sqrt[3]{9}\) (or \(4\times3^{\frac{2}{3}}\))