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Section 1.3 : Radicals
10. Simplify the following expression. Assume that \(x\), \(y\) and \(z\) are positive.
\[\sqrt[4]{{{x^7}{y^{20}}{z^{11}}}}\]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,
\[{x^7}{y^{20}}{z^{11}} = {x^4}{y^{20}}{z^8}{x^3}{z^3} = {x^4}{\left( {{y^5}} \right)^4}{\left( {{z^2}} \right)^4}{x^3}{z^3}\] 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.
\[\sqrt[4]{{{x^7}{y^{20}}{z^{11}}}} = \sqrt[4]{{{x^4}{{\left( {{y^5}} \right)}^4}{{\left( {{z^2}} \right)}^4}{x^3}{z^3}}} = \sqrt[4]{{{x^4}{{\left( {{y^5}} \right)}^4}{{\left( {{z^2}} \right)}^4}}}\,\,\,\sqrt[4]{{{x^3}{z^3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{x\,{y^5}{z^2}\,\,\sqrt[4]{{{x^3}{z^3}}}}}\]