Question

simplify.
\sqrt{45v^{19}}
assume that the variable represents a positive real number.

Explanation:

Step1: Factor the radicand

We can factor \(45\) as \(9\times5\) and \(v^{19}\) as \(v^{18}\times v\) since \(18 + 1=19\) and \(v^{18}=(v^{9})^{2}\). So, \(\sqrt{45v^{19}}=\sqrt{9\times5\times v^{18}\times v}\).

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\)

Applying the property, we get \(\sqrt{9}\times\sqrt{5}\times\sqrt{v^{18}}\times\sqrt{v}\).

Step3: Simplify each square root

We know that \(\sqrt{9} = 3\), \(\sqrt{v^{18}}=v^{9}\) (because \((v^{9})^{2}=v^{18}\)). So substituting these values in, we have \(3\times v^{9}\times\sqrt{5v}\).

Step4: Combine the terms

Combining the terms, we get \(3v^{9}\sqrt{5v}\).

Answer:

\(3v^{9}\sqrt{5v}\)