Question

question 1-10
determine an equivalent form of: (sqrt3{108a^5}
\begin{enumerate}
item (4asqrt{4a^2}
item (3asqrt3{4}
item (36a^6
item (3asqrt3{4a^2}
end{enumerate})

Explanation:

Step1: Factor the radicand

First, we factor \(108a^{5}\) into prime factors and powers of \(a\). We know that \(108 = 27\times4=3^{3}\times4\) and \(a^{5}=a^{3}\times a^{2}\). So, \(\sqrt[3]{108a^{5}}=\sqrt[3]{3^{3}\times4\times a^{3}\times a^{2}}\).

Step2: Apply the cube - root property

The cube - root property states that \(\sqrt[3]{xy}=\sqrt[3]{x}\times\sqrt[3]{y}\) for real numbers \(x\) and \(y\), and \(\sqrt[3]{x^{3}} = x\) for any real number \(x\).
Using the property \(\sqrt[3]{xy}=\sqrt[3]{x}\times\sqrt[3]{y}\), we can rewrite \(\sqrt[3]{3^{3}\times4\times a^{3}\times a^{2}}\) as \(\sqrt[3]{3^{3}}\times\sqrt[3]{a^{3}}\times\sqrt[3]{4a^{2}}\).
Since \(\sqrt[3]{3^{3}} = 3\) and \(\sqrt[3]{a^{3}}=a\), we have \(3\times a\times\sqrt[3]{4a^{2}} = 3a\sqrt[3]{4a^{2}}\).

Answer:

\(3a\sqrt[3]{4a^{2}}\) (corresponding to the option \(3a\sqrt[3]{4a^{2}}\))