Question

simplify.
\sqrt{54v^{16}}
assume that the variable ( v ) represents a positive real number.

Explanation:

Step1: Factor the radicand

We can factor \(54\) as \(9\times6\) and \(v^{16}\) as \((v^{8})^{2}\). So, \(\sqrt{54v^{16}}=\sqrt{9\times6\times(v^{8})^{2}}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a\geq0\) and \(b\geq0\)) and \(\sqrt{a^{2}} = a\) (for \(a\geq0\)), we get:
\(\sqrt{9\times6\times(v^{8})^{2}}=\sqrt{9}\times\sqrt{6}\times\sqrt{(v^{8})^{2}}\)
Since \(\sqrt{9} = 3\) and \(\sqrt{(v^{8})^{2}}=v^{8}\) (because \(v\) is a positive real number), we have:
\(3\times v^{8}\times\sqrt{6}=3v^{8}\sqrt{6}\)

Answer:

\(3v^{8}\sqrt{6}\)