Question

simplify.
\sqrt{8v^{21}}
assume that the variable represents a positive real number.

Explanation:

Step1: Factor the radicand

Factor \(8v^{21}\) into perfect - square factors and non - perfect - square factors. We know that \(8 = 4\times2\) and \(v^{21}=v^{20}\times v\), where \(4 = 2^{2}\) and \(v^{20}=(v^{10})^{2}\) are perfect squares. So, \(\sqrt{8v^{21}}=\sqrt{4\times2\times v^{20}\times v}\).

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a\geq0,b\geq0\))

According to the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\), we can split the square root as follows:
\(\sqrt{4\times2\times v^{20}\times v}=\sqrt{4}\times\sqrt{v^{20}}\times\sqrt{2v}\)

Step3: Simplify the perfect - square roots

We know that \(\sqrt{4} = 2\) and \(\sqrt{v^{20}}=v^{10}\) (since \((v^{10})^{2}=v^{20}\) and \(v\) is a positive real number). So, \(\sqrt{4}\times\sqrt{v^{20}}\times\sqrt{2v}=2\times v^{10}\times\sqrt{2v}\)

Step4: Combine the terms

Combine the terms to get \(2v^{10}\sqrt{2v}\)

Answer:

\(2v^{10}\sqrt{2v}\)