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It has been so long since I have done division inside of radicals that I totally forget the "special rule" for for doing it. -_-

For example, say I wanted to divide the 4 out of this expression:

$\sqrt{1 - 4x^2}$

Is this the right way to go about it?

$\frac{16}{16} \cdot \sqrt{1 - 4x^2}$

$16 \cdot \frac{\sqrt{1 - 4x^2}}{16}$

$16 \cdot \sqrt{\frac{1 - 4x^2}{4}} \Longleftarrow \text{Took the square root of 16 to get it in the radicand as the divisor}$

I know that this really a simple, question. Can't believe that I forgot how to do it. :(

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up vote 3 down vote accepted

There is, in fact, one rather important mistake there. 16 is not the square root of 4. If you replace '16' by'2' in your equation then it is all right. In general, if you divide by everything inside the square root sign by some constant then you should multiply in front by the square root of that constant.

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You say you took the square root of 16 to get it inside the divisor. But taking the square root of 16 and then putting it inside a square root sign is equivalent to taking the square root of that 16 twice, which you definitely don't want to do. If you use 2 instead of 16 then you square 2 to get 4 and then put it into the square root sign to undo the squaring and leave you with the same number. – Donkey_2009 Mar 4 '12 at 23:13
Yes, you're right, according to WolframAlpha:… Could you please explain this a bit better so that it would, perhaps, stick better with me? – Oliver Spryn Mar 4 '12 at 23:14
I see your comment now. Thank you for the additional details! – Oliver Spryn Mar 4 '12 at 23:18
Actually, I understand your comment. :) But if you would like to learn how to make square roots, fractions, powers, etc... you can do that by typing to dollar signs in a row, then between the dollar signs, type in LaTeX code. If you are not familiar with LaTeX, Wikibooks has an article on it: For a quick start, just click the "edit" link underneath my original post, and you can see how I did it. ;) – Oliver Spryn Mar 4 '12 at 23:25
Thanks - I am familiar with LaTeX - just not aware that you could type into MSE using it. I've written a slightly better answer to your question below. – Donkey_2009 Mar 4 '12 at 23:26

Square roots obey the rule $\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}$ . You wanted to take the $4$ out of $\sqrt{1-4x^2}$. $1-4x^2=4\cdot\frac{1-4x^2}{4}$. So $\sqrt{1-4x^2}=\sqrt{4\cdot\frac{1-4x^2}{4}}=\sqrt{4}\cdot\sqrt{\frac{1-4x^2}{4}}=2\cdot\sqrt{\frac{1-4x^2}{4}}$. I think that's the best explanation I can give.

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The correct way to do this, after fixing the mistake pointed out by Donkey_2009, is:

$\dfrac{2}{2} \cdot \sqrt{1-4x^2}$

$= 2 \cdot \dfrac{\sqrt{1-4x^2}}{2}$

$= 2 \cdot \dfrac{\sqrt{1-4x^2}}{\sqrt{4}} \qquad \Leftarrow$ applied $x = \sqrt{x^2}$

$= 2 \cdot \sqrt{\dfrac{1-4x^2}{4}} \qquad \Leftarrow$ applied $\frac{\sqrt a}{\sqrt b} = \sqrt{\frac a b}$

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That really makes everything clear. Thank you! – Oliver Spryn Mar 5 '12 at 1:14

First divide everything inside the radical by what you will want to take out of it, factoring this out while keeping it inside the radical. Then square root what you want to take out of the radical, and take it out.

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