How do you rationalize the denominator of #1/(sqrt(2)+sqrt(3)+sqrt(5))#?
This is a very good question!
If it were simply
and we would multiply by
Let's try something like that and see if it works. (This is what we do with problems of kinds we have not seen before. Try something and see if it works.)
Did that help? (Yes, it did. We now have a more familiar looking problem.
Multiply the numerator if you like, to get:
I've decided to add my version to Jim's as a demonstration or perhaps a warning about the variety of forms the outcome could take:
Consider a simpler problem:
If we were asked to rationalize the denominator of
we would simply multiply both the numerator and denominator by the conjugate of the denominator
In this case
and the result would be
In order to complete the rationalization of the denominator we would need to multiply both the numerator and denominator by
I avoided posting this because I am bothered by the result seems to imply that the sequence of the denominator terms is reflected in the result. This should not be true but I have not (yet) been able to come up with a final result where the terms are interchangeable.
A more general denominator
If we employ the more general denominator of
then we arrive at a numerator of
and a more general (rational) denominator of
The same work as we see for 2, 3, and 5 yields this result.
In the example given, c = a + b, which eliminates the term that would have contained
and a denominator of 2ab.
In the specific instance here we have