Is Taking the Principal Square Root Always Necessary?

[guitarphysics]

I feel kind of ridiculous making this post, but here we go:
What would be the correct answer to this question;
Choose all the number sets (natural, integer, rational, or irrational were the only options given) that
-√81 belongs to, and show how you found your answer.
What I said was this:
-√81=-(plus or minus 9)=-9 or 9.
Therefore, -√81 is a an integer and a rational number.
My teacher (who hates me) said in front of the whole class that I got this wrong. She said I got the final answer right, but that when she asked for the square root of 81, all she was asking for was the principal square root, so my work was incorrect (if I had only taken the principal square root, I of course would have gotten the same number sets, so the final answer was right).
Is it true that when they ask you for the square root of a number, you should only give the principal square root? And if so, do you think she's justified in taking off 1 out of the 2 points that problem was worth?
Thanks

PS. I know how ridiculous it is that I'm in 10th grade and this is what we're doing in math class...

 

[pwsnafu]

We get asked this quite a lot actually.
Yes, only the principle. If $x \geq 0$ then the notation $\sqrt{x}$ is never negative. It part of the definition of the radical symbol √.
This is also why the quadratic formula has a ±.

And if so, do you think she's justified in taking off 1 out of the 2 points that problem was worth?

When teachers create a test, they are supposed to have a marking scheme. So they ask themselves what is the question testing, and assign marks to those elements. But most teachers are lazy in that regard.

Anyway, if it was worth two marks, they would assigned thusly:

1. Understands the theory of square roots, can calculate a simple square root.
2. Understands what the numbers sets are, is able to classify numbers in those sets.

If the point of the question is two test both things, then yes she was justified.

There are a lot of students who like maths because "unlike English, there's only one answer" or "you only need to get the answer right" or something similar. High level maths especially doesn't work like that. Mathematics is using logic to argue something, so you need to justify every step.

 

[guitarphysics]

Thanks a lot for that response! I actually often don't know small details like these because my math education has jumped around from place to place (which is why I feel this is too easy for us now). What they were testing was just the second criterion; understanding number sets and being able to classify numbers in them. We are just now starting radicals, which is why she brought it up during class.

 

[Mentallic]

Yes, the square root operation returns only the principal square root, but there are two solutions to x^2=a which are x=\pm \sqrt{a}. Also notice that if the square root operation returned both the positive and negative root, then we'd have no reason to use the \pm symbol.

As for removing a mark, whether the teacher hates you or not, I feel it was justified. While your answer was correct, you showed a flaw in your Mathematical understanding, and this way, you'll be more likely to learn from your mistake.
Also, I find it odd when markers correct a portion of your working out or answer, but still provide full marks. On the flip side, if she wasn't scrutinizing your work and just gave you full marks, then you still wouldn't have learned of your mistake.

 

[guitarphysics]

Actually, I would have learned about my mistake now :) (Because now this is what we're learning)! But still, you're right, and I usually learn a lot more from mistakes.

 

[Mentallic]

I was making a lot of small mistakes in my working out, and screwing up in various ways during my exams throughout my earlier high school years such that by year 12 - when it really counts - I hardly ran into any serious problems during my exams.

I've done things like thinking my exam was longer than it really was, not realizing there's another booklet to answer or I didn't turn the page to find more questions (this one I did in my school certificate), learning to answer a question the way it expects you to answer it, etc.

And because I was frustrated with making these mistakes, the memory of my mistake stuck in my head more clearly and I more quickly learned to avoid the same mistake in the future. It was a good thing.

 

[guitarphysics]

The same thing happens to me, except more in my personal life than in tests. For example, two years ago when I started becoming interested in science, I did an experiment to see if an amount of water affects how fast that water will evaporate. I had 3 cups with different water levels, and measured how high the water was in each cup (they were transparent). I thought the experiment was well set up and after a while, found that when I started out with less water, it evaporated faster! BUT, something I didn't think about was that the cups had a conical shape (except at the bottom, of course, where the base was flat, not a point), so when I was measuring the changes in height of the waters, I wasn't measuring the change in the volume of water, I was measuring how much the water had gone down in the cup (which was a lot more in the lower parts of the cups, even though less volume had decreased). Because of that, now I am a lot more careful now when setting up experiments! I really hate mistakes when I make them, but they help me a lot in the future :).

 

[Student100]

Haha this also sounds like what happened to me in my precalculus course... The question asked what are the solutions to log(x)+log(x+2)=log(3).. So I put down -3 and 1.. Needless to say the question didn't say to check the solutions so I didn't. I got marked down because -3 isn't in the domain of the function..(it'd be complex and untrue) but when I property of logged it as log(x^2+2x) I went and fought the professor on it. Needless to say he then took the time to educate me on the domain of functions that I didn't quite understand.. Mistakes are always good educational tools!