We use Property (1) to simplify radical expressions by factoring the radicand. For example, to simplify \(\sqrt[3]{108}\text{,}\) we look for perfect cubes that divide evenly into \(108\text{.}\) The easiest way to do this is to try the perfect cubes in order:
\begin{equation*}
1, ~8, ~27, ~64, ~125, \ldots
\end{equation*}
and so on, until we find one that is a factor. For this example, we find that \(108 = 27 \cdot 4\text{.}\) Using Property (1), we write
\begin{equation*}
\sqrt[3]{108}=\sqrt[3]{27}\sqrt[3]{4}
\end{equation*}
Simplify the first factor to find
\begin{equation*}
\sqrt[3]{108}= 3 \sqrt[3]{4}
\end{equation*}
This expression is considered simpler than the original radical because the new radicand, \(4\text{,}\) is smaller than the original, \(108\text{.}\)
We can also simplify radicals containing variables. If the exponent on the variable is a multiple of the index, we can extract the variable from the radical. For instance,
\begin{equation*}
\sqrt[3]{12}=x^{12/3}=x^4
\end{equation*}
(You can verify this by noting that \((x^4)^3 = x^{12}\text{.}\)) If the exponent on the variable is not a multiple of the index, we factor out the highest power that is a multiple. For example,
\begin{align*}
\sqrt[3]{x^{11}} \amp = \sqrt[3]{x^9 \cdot x^2}\amp\amp\blert{\text{Apply Property (1).}}\\
\amp = \sqrt[3]{x^9}\cdot\sqrt[3]{x^2} \amp\amp\blert{\text{Simplify }\sqrt[3] {x^9}=x^{9/3}.} \\
\amp = x^3 \sqrt[3]{x^2}
\end{align*}
To simplify roots of fractions, we use Property (2), which allows us to write the expression as a quotient of two radicals.
We can also use Properties (1) and (2) to simplify products and quotients of radicals.
