Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 1.3 : Radicals
11. Simplify the following expression. Assume that \(x\), \(y\) and \(z\) are positive.
\[\sqrt[3]{{54{x^6}{y^7}{z^2}}}\]Show All Steps Hide All Steps
Start SolutionRecall that by simplify we mean we want to put the expression in simplified radical form (which we defined in the notes for this section).
To do this for this expression we’ll need to write the radicand as,
\[54{x^6}{y^7}{z^2} = \left( {27{x^6}{y^6}} \right)\left( {2{y^1}{z^2}} \right) = {3^3}{\left( {{x^2}} \right)^3}{\left( {{y^2}} \right)^3}\left( {2y{z^2}} \right)\] Show Step 2Now that we’ve gotten the radicand rewritten it’s easy to deal with the radical and get the expression in simplified radical form.
\[\sqrt[3]{{54{x^6}{y^7}{z^2}}} = \sqrt[3]{{{3^3}{{\left( {{x^2}} \right)}^3}{{\left( {{y^2}} \right)}^3}\left( {2y{z^2}} \right)}} = \sqrt[3]{{{3^3}{{\left( {{x^2}} \right)}^3}{{\left( {{y^2}} \right)}^3}}}\sqrt[3]{{2y{z^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{3{x^2}{y^2}\,\,\sqrt[3]{{2y{z^2}}}}}\]