Question

write the following in simplified radical form.
\sqrt4{162}

Explanation:

Step1: Factor 162

First, we factor 162. We know that \(162 = 81\times2\), and \(81 = 3^4\). So, \(162 = 3^4\times2\).

Step2: Simplify the radical

Now, we have \(\sqrt[4]{162}=\sqrt[4]{3^4\times2}\). Using the property of radicals \(\sqrt[n]{ab}=\sqrt[n]{a}\times\sqrt[n]{b}\) (where \(a\geq0\), \(b\geq0\) and \(n\) is a positive integer), we can split the radical: \(\sqrt[4]{3^4\times2}=\sqrt[4]{3^4}\times\sqrt[4]{2}\).
Since \(\sqrt[4]{3^4}=3\) (because the fourth root of \(3^4\) is 3, as \((3)^4 = 3^4\)), we get \(3\times\sqrt[4]{2}=3\sqrt[4]{2}\).

Answer:

\(3\sqrt[4]{2}\)