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What would the result of $4\sqrt x$ multiplied by $18x^2-12$? Would it just be $18x^{10/4}$?

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Do you mean $4\sqrt x(18x^2-12)$? – Brian M. Scott Jan 24 '13 at 2:43
Yes that's what I meant :) – Captn Buzz Jan 24 '13 at 2:45

If by simplify you mean multiply out,

$$4\sqrt x\left(18x^2-12\right)=4x^{1/2}\left(18x^2-12\right)=72x^{5/2}-48x^{1/2}\;,$$

If, on the other hand, you mean factor as completely as possible,

$$4\sqrt x\left(18x^2-12\right)=24\sqrt x\left(3x^2-2\right)=24\sqrt x\left(\sqrt3 x-\sqrt2\right)\left(\sqrt 3x+\sqrt2\right)\;.$$

There is no way to reduce it to a single term with a single power of $x$.

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$$4\sqrt x(18x^2-12)=\begin{cases}72\sqrt x\,x^2-48\sqrt x=72x^{5/2}-48x^{1/2}\\{}\\24\sqrt x(3x^2-2)\end{cases}$$ no: it is not what you thought it is.

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WolframAlpha shows the same. The roots of your expression are $0$ and $+/-$ $\sqrt{2/3}$

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