Question: Finding the n^th root of a perfect n^th power monomial Simplify. √243u^25 Assume that the variable…

Finding the n^th root of a perfect n^th power monomial Simplify. √243u^25 Assume that the variable represents a positive real number.

Solution

To simplify the expression \(\sqrt[5]{243u^{25}}\), we can break it into two parts: \(\sqrt[5]{243}\) and \(\sqrt[5]{u^{25}}\). First, find \(\sqrt[5]{243}\). Notice that \(243 = 3^5\). \[ \sqrt[5]{243} = \sqrt[5]{3^5} = 3 \] Next, find \(\sqrt[5]{u^{25}}\). \[ \sqrt[5]{u^{25}} = (u^{25})^{1/5} = u^{25 \cdot \frac{1}{5}} = u^5 \] Combine both results: \[ \sqrt[5]{243u^{25}} = 3 \cdot u^5 = 3u^5 \] The simplified expression is \(3u^5\).