From Surds, to Ab-Surds

I still use the word surd for irrational square roots, and I know there is a undercurrent in modern math education to remove what is considered "difficult"  language in the classroom.  I leave that to those still fighting in the classroom to decide, but as a historian, the term is too rich in content not to use it, and teach it. Hence the title of From Surds, to Ab-Surds


When I first saw the image above I thought, Oh, that's neat. I mean I know it doesn't normally work, but then I also like the crazy wrong cancellations that work, such as

As I sat and tried to think of other similar "wrong" examples that work with surds, I realized it might make a really good first or second year algebra challenge.  There is nothing very difficult about the algebra itself, so it allows the problem to be setting up the algebraic structure of the arithmetic problem.  

My early thoughts quickly generated enough to recognize a pattern to generate as many as I would want, *** 

 \( \sqrt {3 \frac{3}{8}} = 3 \sqrt{ \frac{3}{8}} \) 

*** or one in higher values.  For example:

 \( \sqrt {49 \frac{49}{2400}} = 49 \sqrt{ \frac{49}{2400}} \) 

  And in general it will always work in this form::

    \( \sqrt {n \frac{n}{n^2-1}} = n \sqrt{ \frac{n}{n^2-1}} \)