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Section 1.3 : Radicals

15. Multiply the following expression. Assume that \(x\) is positive.

\[\left( {\sqrt[3]{x} + 2\,\,\sqrt[3]{{{x^2}}}} \right)\left( {4 - \sqrt[3]{{{x^2}}}} \right)\] Show Solution

All we need to do here is do the multiplication so here is that.

\[\begin{align*}\left( {\sqrt[3]{x} + 2\,\,\sqrt[3]{{{x^2}}}} \right)\left( {4 - \sqrt[3]{{{x^2}}}} \right) & = 4\sqrt[3]{x} - \sqrt[3]{x}\,\,\sqrt[3]{{{x^2}}} + 8\,\,\sqrt[3]{{{x^2}}} - 2\,\,\sqrt[3]{{{x^2}}}\,\,\sqrt[3]{{{x^2}}}\\ & = 4\sqrt[3]{x} - \sqrt[3]{{{x^3}}} + 8\,\,\sqrt[3]{{{x^2}}} - 2\,\,\sqrt[3]{{{x^4}}}\\ & = 4\sqrt[3]{x} - \sqrt[3]{{{x^3}}} + 8\,\,\sqrt[3]{{{x^2}}} - 2\,\,\sqrt[3]{{{x^3}}}\,\,\sqrt[3]{x}\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{4\sqrt[3]{x} - x + 8\,\,\sqrt[3]{{{x^2}}} - 2x\,\,\sqrt[3]{x}}}\end{align*}\]

Don’t forget to simplify any resulting roots that can be. That is an often missed part of these problems and when dealing with roots other than square roots there can be quite a bit of work in the simplification process as we saw with this problem.