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Section 1.3 : Radicals

11. Simplify the following expression. Assume that \(x\), \(y\) and \(z\) are positive.


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Recall 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 2

Now that we’ve gotten the radicand rewritten it’s easy to deal with the radical and get the expression in simplified radical form.

\[\sqrt[3]{{54{x^6}{y^7}{z^2}}} = \sqrt[3]{{{3^3}{{\left( {{x^2}} \right)}^3}{{\left( {{y^2}} \right)}^3}\left( {2y{z^2}} \right)}} = \sqrt[3]{{{3^3}{{\left( {{x^2}} \right)}^3}{{\left( {{y^2}} \right)}^3}}}\sqrt[3]{{2y{z^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{3{x^2}{y^2}\,\,\sqrt[3]{{2y{z^2}}}}}\]