Need help on my review sheet

pka

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\(\displaystyle \frac{1+t+t^2}{\sqrt t}=\frac{(1+t+t^2)\sqrt t}{t}=~?\)
 

Dr.Peterson

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I am not sure where to start on the following problem:

Simplify the following: \(\displaystyle \frac{1+t+t^2}{\sqrt{t}}\)
You start with what you have, and do whatever you can do -- which isn't much! In fact, the "simplified" form won't look very simple, so the hard part really is just to decide that there's nothing more to do!
 

Sely

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\(\displaystyle \frac{1+t+t^2}{\sqrt t}=\frac{(1+t+t^2)\sqrt t}{t}=~?\)
Thank you! I wasn’t quite sure if I could take it further than this, so that had me a little confused.
 

Jomo

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Thank you! I wasn’t quite sure if I could take it further than this, so that had me a little confused.
Did you try to factor t^2 + t + 1??
 

Sely

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Dr.Peterson

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You can't (at least not in a way that makes anything simpler).

One problem here is that "simplicity is in the eye of the beholder" (unless the next thing I planned to do with it required a different form). I would personally leave it just as it was given to you; but many teachers require rationalizing the denominator. Beyond that, if the numerator and denominator had a common factor, you would cancel it, but they don't. In fact, if there were a common factor, it would have to be t, so you can easily see that factoring will not help.
 

pka

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I didn’t know I could..?
Frankly I have no idea what answer is expected!
If I had set the question this would have been my answer: \(\displaystyle \frac{1+t+t^2}{\sqrt t}=t^{-1/2}+t^{1/2}+t^{3/2}\)
How in the world, given such vague instruction can anyone know the expected answer?
 

Dr.Peterson

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This is a good example where context is needed. Within a given class at this level, there will often be a standard for what is considered "simple" (for a particular kind of expression); that makes it easy for both teacher and students to know what is expected.

In real life, you put it in whatever form is most useful for what comes next. In this case, unfortunately, what comes next is turning it in to the teacher for judgment, and the teacher gets to decide based on class rules, written or unwritten.

So, @Sely, what have you been taught about what is "simple enough"?
 

Otis

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\(\displaystyle \frac{1+t+t^2}{\sqrt t}=\frac{(1+t+t^2)\sqrt t}{t}\)
I think that's good enough (rationalizing the denominator). The image shows part of a review.

\(\;\)
 

Sely

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This is a good example where context is needed. Within a given class at this level, there will often be a standard for what is considered "simple" (for a particular kind of expression); that makes it easy for both teacher and students to know what is expected.

In real life, you put it in whatever form is most useful for what comes next. In this case, unfortunately, what comes next is turning it in to the teacher for judgment, and the teacher gets to decide based on class rules, written or unwritten.

So, @Sely, what have you been taught about what is "simple enough"?
My teacher has never really explained what is considered “simple enough”. I don’t know when I have done enough and can move on
 

Jomo

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I didn’t know I could..?
You can't. I am just making sure that you tried just in case it could be factored.
 

Otis

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My teacher has never really explained what is considered “simple enough”. I don’t know when I have done enough and can move on
Hi Sely. That review sheet seems to cover some basics from intermediate algebra, trigonometry and precalculus. If you just started a calculus course, for example, then the purpose of giving students such review exercises could be a desire of the instructor to get a feel for how many students are ready.

I wouldn't be too concerned about "simple enough". The point seems to be whether you remember some basics (eg: algebra steps for converting a complex ratio into a single ratio; rationalizing a denominator; basic trig identities; switching from radical notation to exponential notation).

As you've seen, more than one answer is possible. (I think that's true for all four exercises, on that sheet.) Any valid result that demonstrates your knowledge of some basic, prerequisite material is probably good enough.

😎
 
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