# What would this equation be simplified to?

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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$$4\sqrt x(18x^2-12)=\begin{cases}72\sqrt x\,x^2-48\sqrt x=72x^{5/2}-48x^{1/2}\\{}\\24\sqrt x(3x^2-2)\end{cases}$$

...so 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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$$4\sqrt x(18x^2-12) = 4x^{1/2}(18x^2 - 12)$$ $$=72 x^{1/2}x^2 - 48 x^{1/2}\tag{distribute}$$ $$=72x^{(1/2) + 2} - 48x^{1/2} \tag{add exponents}$$ $$= 72x^{5/2} - 48 x^{1/2}\tag{expanded form}$$
$$= 24x^{1/2}(3x^2 - 2) = 24\sqrt{x}(3x^2 - 2)\quad\quad\quad\tag{factor out common terms}$$ or $$24\sqrt{x}(3x^2 - 2) = 24\sqrt{x}(\sqrt{3}x - \sqrt{2})(\sqrt{3}x + \sqrt{2})\quad\quad\quad\quad\quad\quad\quad\tag{completely factored form}$$
From the completely factored form, one can see that the roots of your expression (the values of $x$ at which your expression is equal to $0$) are $$x_1 = 0,\;x_2 = \sqrt{\frac{2}{3}},\;x_3 = - \sqrt{\frac{2}{3}}$$
So, clearly, your expression cannot "just be" $18x^{10/4} = 18x^{5/2}$