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Section 1.3 : Radicals

17. Rationalize the denominator. Assume that \(x\) is positive.

\[\frac{9}{{\sqrt[3]{{2x}}}}\] Show Solution

For this problem we need to multiply the numerator and denominator by \(\sqrt[3]{{{{\left( {2x} \right)}^2}}}\) in order to rationalize the denominator.

\[\frac{9}{{\sqrt[3]{{2x}}}} = \frac{9}{{\sqrt[3]{{2x}}}}\frac{{\sqrt[3]{{{{\left( {2x} \right)}^2}}}}}{{\sqrt[3]{{{{\left( {2x} \right)}^2}}}}} = \frac{{9\,\,\sqrt[3]{{{{\left( {2x} \right)}^2}}}}}{{\sqrt[3]{{{{\left( {2x} \right)}^3}}}}} = \frac{{9\,\,\sqrt[3]{{{{\left( {2x} \right)}^2}}}}}{{2x}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{9\,\,\sqrt[3]{{4{x^2}}}}}{{2x}}}}\]