Always rationalize the denominator?

[Ebba Sen Pai]

I have two question:
1) Is it always necessary to rationalize the denominator of a fraction (even if you are not explicitly told do do so, but instead simply to "Factor the expression completely."
(I was of the mind it is, but in a video from my pre-calc class, the instructions do not seem to view the exponential expression(s) in the denominator as an issue for their solution).

2)If it is not, are there certain keywords I should look for to determine when it is, and is not required to rationalize a denominator (outside of beign told explicitly to do so)?

As far as I am aware, the exponent is 4/3 in the denominator is just a secret radical lying in the wait.
As always, I thoroughly appreciate your help (everyone). I wouldn't have made it to Pre-Calc without this amazing site.

 

[ksdhart2]

I'd be reticent to suggest that you always have to follow any procedure. Strictly speaking, rationalizing the denominator isn't necessary, because the expression has the same value regardless. Many people think doing so makes the expression looker "simpler" or "cleaner" but I'm not one of them. That's just my personal preference, though.

As far as knowing when you might want to rationalize the denominator, I'm not sure of any specific keywords to look for in the problem text. However, you'll quickly get a feel for if your teacher wants you to do it.

 

[Dr.Peterson]

Rationalizing the denominator is only sometimes appropriate. I would only do it without being told when a teacher has stated that it is always expected; you can certainly ask your teacher about such a policy. In some cases, the rationalized-denominator form is considerably less simple than otherwise.

But in this case, I would say that when a problem is given in exponential rather than radical form, the answer is likely expected in the same form, and rationalizing is typically done only in radical form.

 

[Ebba Sen Pai]

I'd be reticent to suggest that you always have to follow any procedure. Strictly speaking, rationalizing the denominator isn't necessary, because the expression has the same value regardless. Many people think doing so makes the expression looker "simpler" or "cleaner" but I'm not one of them. That's just my personal preference, though.

As far as knowing when you might want to rationalize the denominator, I'm not sure of any specific keywords to look for in the problem text. However, you'll quickly get a feel for if your teacher wants you to do it.

This helps enormously! Thanks.

 

[Ebba Sen Pai]

Rationalizing the denominator is only sometimes appropriate. I would only do it without being told when a teacher has stated that it is always expected; you can certainly ask your teacher about such a policy. In some cases, the rationalized-denominator form is considerably less simple than otherwise.

But in this case, I would say that when a problem is given in exponential rather than radical form, the answer is likely expected in the same form, and rationalizing is typically done only in radical form.

This makes perfect sense. I just got a new instructor, so this new expectation is different, while I think I subliminally internalised the expectations of always rationalising from my prior instructor. Thanks!

 

[JeffM]

The practice of rationalizing denominators has, I believe, two roots. One is that expressing fractions in a standard form makes it easier to compare fractions. In that respect, it makes no mathematical difference what method is used to express a fraction, but communication is facilitated by using one method consistently.

The other root is I believe no longer of practical importance, but if you needed to compute

[MATH]\dfrac{1}{\sqrt{3}}[/MATH]
by hand, you would find it far easier to divide

[MATH]\dfrac{\sqrt{3}}{3}[/MATH]
by hand.