fraction and square root manipulation issue

i was solving a calculus problem and i yielded the answers 1/sqrt(5) and 2/sqrt(5). when i looked at the solution in the book it showed the answer as (1/5)*sqrt(5) and (2/5)*sqrt(5). I checked it with my calculator and it shows that my answers and the book's answers are equal, but I cannot figure out how to transform 1/sqrt(5) and 2/sqrt(5) to look like (1/5)*sqrt(5) and (2/5)*sqrt(5). Can anyone show me how to do this? Also what is a better way for me to express the numbers here on this forum?

Re: fraction and square root manipulation issue

$\dfrac 1 {\sqrt{5}}=\dfrac 1 {\sqrt{5}} \dfrac {\sqrt{5}}{\sqrt{5}}=\dfrac{\sqrt{5}}{5}$

Re: fraction and square root manipulation issue

Yes, they are equal, but by convention if you have a square root in the denominator you perform the cionversion as romsek showed so that you end up with the square root in the numerator instead. I guess at some point someone just decided that the number $\displaystyle \frac {\sqrt 5}5 $ "looks better" than $\displaystyle \frac 1 {\sqrt 5}$, and we've been doing it this way ever since.

Re: fraction and square root manipulation issue

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Originally Posted by

**ebaines** Yes, they are equal, but by convention if you have a square root in the denominator you perform the cionversion as romsek showed so that you end up with the square root in the numerator instead. I guess at some point someone just decided that the number $\displaystyle \frac {\sqrt 5}5 $ "looks better" than $\displaystyle \frac 1 {\sqrt 5}$, and we've been doing it this way ever since.

Back when I used to comment at a different site that will not be named, I gave a response like this. It was kindly explained to me that, back when we used to compute things with pencil and paper, dividing by integers was easier than dividing by an approximation to an irrational. So I doubt it was aesthetics; more likely, it was simply convenience. And now it is convention.

Re: fraction and square root manipulation issue

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Originally Posted by

**catenary** Also what is a better way for me to express the numbers here on this forum?

This site has a program called LaTeX that renders math expressions in a lovely format. I personally do not advise students to learn it unless they plan to ask a lot of questions because it is a fussy sort of program and students have lots of things to learn. Nevertheless, lots of people think my advice sucks. There are websites that explain LaTeX, but probably the easiest way to start learning it is to reply with quote to posts on this site that have LaTeX expressions so you can see the actual code used to invoke LaTeX and to use it. After you have the basic idea, then the websites can be used to fill in the gaps.

Re: fraction and square root manipulation issue

There is a bit more to it than just "convention" or "when we used to compute things with paper and pencil". When you are combining **algebraic** formulas it is simpler if the radicals are only in the numerator.

Re: fraction and square root manipulation issue

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Originally Posted by

**HallsofIvy** There is a bit more to it than just "convention" or "when we used to compute things with paper and pencil". When you are combining **algebraic** formulas it is simpler if the radicals are only in the numerator.

Thanks. Makes sense because who wants to find common denominators with a bunch of radicals in them. So it is a convention that is still convenient even in the day of calculators. Next time, I'll give a better answer than the one I learned from mrspi.

Re: fraction and square root manipulation issue

Thanks guys for explaining it. Actually I used to know this and use this a lot, I even had teachers that wanted the answers to the tests written in the format where there is no nth root in the denominator, but lack of using it just made me completely forget all about it and I really couldn't remember this anymore until now. thanks again!