Question

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

Explanation:

Step1: Factor the radicand

We can factor \(8u^{21}\) as \(4\times2\times u^{20}\times u\). So, \(\sqrt{8u^{21}}=\sqrt{4\times2\times u^{20}\times u}\).

Step2: Use the property of square roots

Recall that \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)) and \(\sqrt{a^{n}} = a^{\frac{n}{2}}\) when \(a\geq0\) and \(n\) is an integer.
First, \(\sqrt{4}=2\), \(\sqrt{u^{20}}=u^{10}\) (since \(\sqrt{u^{20}}=(u^{20})^{\frac{1}{2}} = u^{10}\) as \(u\) is positive). Then we have:
\(\sqrt{4\times2\times u^{20}\times u}=\sqrt{4}\times\sqrt{u^{20}}\times\sqrt{2u}=2\times u^{10}\times\sqrt{2u}\)

Step3: Simplify the expression

Combining the terms, we get \(2u^{10}\sqrt{2u}\).

Answer:

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