I can't find any decent information on dealing with square roots. I know that sq(X) . sq(Y) = sq(XY) and conversely sq(X) / sq(Y) = sq(X/Y) but thats all the tutorials tell you.

If I have a question like this for instance "X . sq(Y)" the answer is not sq(XY). I gave that some thought and heres the method I came up with "(X . Y)^2 = Z". For example "(3 . sq(9))^2 = 9". I just get rid of the square root sign over Y then multiply X by expanded Y and multiplied the result by itself. Is this the commonly used method for multiplying square roots by normal numbers or is there an easier method?

Anyhow my method for that works so I don't really care about that heres the one I can't figure out. I do the method above backwards but theres probably a simpler method.

Q.) Say I have sq(81) but I'm asked to write that as Asq(3). A being the coefficient of sq(3).

I was asked that on a test and I didn't have time to work it out in my head.

x*sqrt(y)=sqrt(x^2*y)

I think thats what you're asking. In your example, you could use that method this way:

x*sqrt(3)=sqrt(81)
sqrt(x^2*3)=sqrt(81)
x^2*3=81
x^2=27
x=5.19615242

But really thats nothing more of a simple retooling of your original principles.

also, the method you came up with won't work as far as I can tell.

(3 . sq(9))^2=81, not 9.

Thanks that explains it. BTW I meant my answer comes out to sq(81) so its the right answer but your methods a little easier so I'll memorize that instead.

Don't memorize it. Just realize that the way I posted is just a different way of showing the same basic rules of square roots you already know.

If a=x^2 then

sqrt(ay)=sqrt(a)*sqrt(y)=x*sqrt(y)

I just did the process in reverse:

x*sqrt(y)=sqrt(x^2)*sqrt(y)=sqrt(x^2*y)