Can radicals be left in the denominator in the case of a negative exponent?

[CactUs]

That is, which is the correct simplified form of x^-5/2? Is it [MATH]\frac{1}{\sqrt{x^{5}}}[/MATH] or [MATH]\frac{\sqrt{x^{5}}}{x^{5}}[/MATH]? Does canceling out the common factor x take precedence, or is getting rid of the radical more important?

Thanks so much!

 

[Deleted member 4993]

That is, which is the correct simplified form of x^-5/2? Is it [MATH]\frac{1}{\sqrt{x^{5}}}[/MATH] or [MATH]\frac{\sqrt{x^{5}}}{x^{5}}[/MATH]? Does canceling out the common factor x take precedence, or is getting rid of the radical more important? Thanks so much!

In my opinion, among all those choices, the most useful expression is x^-5/2.

It depends on the "whim" of the instructor (or grader).

 

[Dr.Peterson]

Can radicals be left in the denominator in the case of a negative exponent?

That is, which is the correct simplified form of x^-5/2? Is it [MATH]\frac{1}{\sqrt{x^{5}}}[/MATH] or [MATH]\frac{\sqrt{x^{5}}}{x^{5}}[/MATH]? Does canceling out the common factor x take precedence, or is getting rid of the radical more important?

Thanks so much!

As I have often said, "Simplification is in the eye of the beholder".

Ultimately, what to consider simplest depends on what you are going to do next -- add it, or multiply it, or graph it, or evaluate it by hand, or compare it to something else, for example. If what you are going to do next is to submit it to a teacher, then the teacher gets to decide (and must tell you) what their rules are -- or that it doesn't matter. If it is to check your answer in the back of the book, then it will be easiest if you have observed what form they tend to leave answers in (though you can also convert one to the other as a check, or use a calculator).

There is no one correct form.

 

[Steven G]

I believe that final answers should not have radicals in the denominator. For intermediate steps it is fine to have a radical in the denominator.

As others have said, it really depends on the grader. If I made a big deal in class about not leaving radicals in the denominator for a final answer then you would lose points on an exam if you did.

Where do you see the common factor of x? Please post back with the answer.

 

[yoscar04]

I recall being told many years ago that it is better to not have radicals in the denominator when performing numerical evaluations. But I am not sure how bad it is if the radicals are kept in the denominator.

 

[Dr.Peterson]

I recall being told many years ago that it is better to not have radicals in the denominator when performing numerical evaluations. But I am not sure how bad it is if the radicals are kept in the denominator.

That basically applies to numbers, rather than to variables, and comes from the days when you would have to choose between dividing (by hand) 1 by [MATH]\sqrt{2}[/MATH], or [MATH]\sqrt{2}[/MATH] by 2. The latter is far easier! (Try it. And then try again with more decimal places.)

Today, I think the main reason it is commonly considered standard to rationalize denominators is so that there is a standard form to compare students' answers to. It's also needed when you add two radical expressions together. But there are many cases where the rationalized form of an expression is far less simple, and I would never do it unless I had some specific reason to.

 

[yoscar04]

That basically applies to numbers, rather than to variables, and comes from the days when you would have to choose between dividing (by hand) 1 by [MATH]\sqrt{2}[/MATH], or [MATH]\sqrt{2}[/MATH] by 2. The latter is far easier! (Try it. And then try again with more decimal places.)

Today, I think the main reason it is commonly considered standard to rationalize denominators is so that there is a standard form to compare students' answers to. It's also needed when you add two radical expressions together. But there are many cases where the rationalized form of an expression is far less simple, and I would never do it unless I had some specific reason to.

Yes, my comment was about having radicals (numerical) in the denominator . Actually, now I think that another reason is that it is easier to add fractions without radicals in the denominator than with them.

 

[yoscar04]

In the spirit of Dr.Peterson comment, there are many expressions in Physics where we keep the radicals in the denominator, i.e.



which is standard practice when computing things in QM Another example:



and nobody bothers in rationalizing those expressions.