Simplifying an expression?

[Dav23]

This problem has been troubling me for a while but i still cannot figure it out

3-√2
÷
(2√3)+5

 

[Dav23]

yes that is the problem and I cannot show my steps as I have no idea where I should start

 

[Mrspi]

A common way to start simplifying fractions that involve radicals in the denominator is to eliminate them. Quite frankly, I am not sure why that is considered such a universally great improvement, but it is standard and frequently does help in simplification.

Back in the days BC (before calculators) when computations were done largely by hand, one would definitely appreciate the difference between doing a division with a radical in the denominator, and doing an equivalent division with the denominator being a whole number!

For example, consider doing the division 5/sqrt(2), which equates to 5 / 1.414214. Not something I would want to do by long division!

However, if you rationalize the denominator (multiply numerator and denominator by sqrt(2)) to get 5 sqrt(2) / 2, you're faced with 5(1.414214) / 2....that's LOTS easier, in my opinion.

Of course, we now have calculators readily available, (and as a result, kids who absolutely cannot do something like 58/7!!!) so it really doesn't make much difference if you're dividing by a radical or not. However, old habits die hard, I guess.

 

[HallsofIvy]

Some people have asked, perhaps on this forum, 'why do we always rationalize the denominators?' The only correct is "we don't, always". Sometimes it helps to rationalize the numerators. It depends upon the problem. It is generally inconvenient to have radicals in both numerator and denominator. And, since the most common operation with fractions is to add them- for which we need common denominators- it is probably most convenient to have integer denominators which we can do as long as there are not radicals in the denominator.

 

[Deleted member 4993]

Jeff,

Watch out!! Thank-you Nazi will get you.....

By the way, a similar process (of multiplying with conjugates) is applied for complex number division also - but that is another topic....