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Can you help me figure out this problem? - A Sorta Fairytale
October 2013
 
 
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hymnia
hymnia
Joie
Wed, Mar. 25th, 2009 07:11 pm
Can you help me figure out this problem?

It's driving me crazy. I was working with my Geometry students today, and I got stuck on a problem. We were on the last step. I knew what the answer was because I had the teacher's guide, but I couldn't for the life of me figure out how it went from the previous step to the final answer. Even after looking up simplifying radicals in a couple of different textbooks, and trying to find the info with a Google search, I'm still stumped. I thought perhaps some of the smart people on my flist could explain it to me. :D

Here is the problem:
Find the geometric mean of 4√3 and 5√2.

According to my teacher's guide, you set up the proportion, cross multiply, and isolate the variable. At that point you get:
x=√(4√3 * 5√2) <--All that is under the first radical.

Now, I can think of one way to deal with "square root of square root of…": change to fractional exponents. But that only makes the problem messier, as far as I can tell. I can also use the calculator to approximate square roots, which gives the answer of around 6.98. But my answer key says that the final answer is the whole number 7, so there must be a better way to simplify it. Do you know what it is?

Joie

Tags:
Current Mood: confused confused

24CommentReplyShare

blpurdom
blpurdom
Prehumous Professor of Morbid Bibliomancy
Wed, Mar. 25th, 2009 11:49 pm (UTC)

I've run across more than one math textbook, for both me and my kids, with typos in it, so that's one possibility. However, since the answer of 7 is so very close to 6.98, is there a possibility that the question says to round the answer up to the nearest whole number and in trying to work this out you (quite understandably) overlooked that? It's also possible that someone did mean to include that verbiage in the question and in the editing process this got overlooked.


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hymnia
hymnia
Joie
Thu, Mar. 26th, 2009 01:21 am (UTC)

Oh, good point. I'll have to double-check the book tomorrow to see if it said to round. (And if it didn't, it's certainly possible they meant it to. It wouldn't be the first mistake we've found in this book.)


ReplyThread Parent
alphabet26
alphabet26
that kind of girl
Wed, Mar. 25th, 2009 11:54 pm (UTC)
I'm no help

As soon as I saw "geometry," I'm afraid my eyes glazed over. Algebra, even trig, I liked okay. Geometry and I never got on, though.


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hymnia
hymnia
Joie
Thu, Mar. 26th, 2009 01:23 am (UTC)
Re: I'm no help

But this *is* algebra. I mean, geometric mean is geometry, sure, but simplifying a radical, which is where I was stuck on the problem, is algebra. :P


ReplyThread Parent
rabbithabit12
h
Thu, Mar. 26th, 2009 12:25 am (UTC)

i have no idea, i'm only in eigth grade after all....


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hymnia
hymnia
Joie
Thu, Mar. 26th, 2009 01:25 am (UTC)
Soifon icon!

True, this is more of a 9th/10th grade concept. Maybe you could try again in a couple years? (j/k)


ReplyThread Parent
major_dallas
major_dallas
Nate the Great
Thu, Mar. 26th, 2009 12:42 am (UTC)

ok as its been ages since my mathing days, the little check-mark figures are supposed to mean square root right? so the problem you want to find out is the Geometric Mean of (4 X the square root of 3) and (5 X square root of 2) and how the answer would be 7?


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hymnia
hymnia
Joie
Thu, Mar. 26th, 2009 01:19 am (UTC)

Yep, that's right.


ReplyThread Parent
major_dallas
major_dallas
Nate the Great
Thu, Mar. 26th, 2009 01:59 am (UTC)
using Gilderoy just in case I did blow the math...

ok... so using a calculator on my PC, I get 6.9992710231611664577286727297385...

For this explanation, "X" = multiplying ;) and SR = Square Root and I did not do any rounding on the Square Roots

I got this by first, getting the SR of 3 and X by 4(6.9282032302755091741097853660235) than getting the SR of 2 and X by 5(7.0710678118654752440084436210485) then X those 2 sums which gave me 48.989794855663561963945681494114, then using the SR function one last time, got the above number...

since a PC calculator can go up to like almost 15 digits behind a decimal point, it can be fairly more accurate with SR numbers which is what I think happened here...

as to writing it all out...uhhmm, its difficult without a piece of paper, pencil and a calculator that is supplied by a computer...


ETA: Oh...someone already this part...oops

Edited at 2009-03-26 02:03 am (UTC)


ReplyThread Parent
hymnia
hymnia
Joie
Thu, Mar. 26th, 2009 09:20 pm (UTC)
Re: using Gilderoy just in case I did blow the math...

I think we've got it figured out now. Turns out it was a rounding thing after all. I still need to double-check the instructions on the problem to see if they said anything about rounding. (If not, shame on them, IMO.) In any case, thanks for your input. :)


ReplyThread Parent
liriop
liriop
liriop
Thu, Mar. 26th, 2009 01:25 am (UTC)

I think that the problem is that you approximated the square roots. In my calculator, without approximating anything (besides the approximation that the calculator has to do because they are irrational numbers), the answer was 6,999271023161, which could be approximated to 7, but never to 6,98.


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liriop
liriop
liriop
Thu, Mar. 26th, 2009 01:26 am (UTC)

Sorry I wrote the numbers the Brazilian way (commas instead of dots). That should be 6.999271023161 and 6.98.


ReplyThread Parent
hymnia
hymnia
Joie
Thu, Mar. 26th, 2009 01:32 am (UTC)

Yeah, but the fact that it's coming out so close to 7 suggests to me that there might be a way to simplify it without resorting to even very close approximations, like those on your calculator.


ReplyThread Parent
springdove
springdove
Kristi
Thu, Mar. 26th, 2009 02:07 am (UTC)

It's been so long since I've done math that I'm a little rusty, but the only way I can think of to do the problem leaves me thinking it would be almost 7, rather than 7 on the dot. I think the rounding suggestion makes best sense.

I hated geometry, too, incidentally. I will also admit that it has been so long for me that I had no idea even how to set up the proportion. :P


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hymnia
hymnia
Joie
Thu, Mar. 26th, 2009 09:16 pm (UTC)

Yeah, it looks like that's the right explanation. (BTW, I <3 your icon!)


ReplyThread Parent
peachespig
peachespig
peachespig
Thu, Mar. 26th, 2009 02:50 am (UTC)

major_dallas and liriop already gave you the decimal expansion to more significant digits than anyone is likely to need, I agree with their results (at least to the first 6 digits, which is all I bothered to check).

This number is definitely close to 7, but it is not precisely 7; it agrees with 7 to three significant digits but not four. If the teaching manual says it's exactly 7, it's lying. I don't really think you can simplify it much; you could write it as √(20√6) I suppose, or even 2√(5√6), but none of those are particularly illuminating. It is without a doubt an irrational number.

Here's a cute way to see how close it is to 7 without being 7. The number inside the square root is 4√3 times 5√2. Now 4=√16, so we can write the first number as √16√3 = √48. Also 5=√25, so the second number is √25√2 = √50. So multiplying those together gives √48√50 = √2400. Your entire number is the square root of that, so it's √√2400, or 2400 to the power 1/4.

What about seven exactly? Seven squared is 49, and 49 squared is 2401. So 7 to the fourth power is 2401; meaning 2401 to the power 1/4 is exactly 7. Your number is 2400 to the power 1/4; just a tiny bit less.


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hymnia
hymnia
Joie
Thu, Mar. 26th, 2009 09:15 pm (UTC)

Okay, I'm now totally laughing at myself because at one point in my messing around with this problem I actually got 2400^(1/4), and I thought, "But I don't have a scientific calculator so I can check to see if that comes out to 7." ajfdk;ajfdk;afjdkla Um, duh Joie. Just take 7 and raise it to the fourth power and see if it matches. ajd;kafdjalfdjsa Oops.

So anyway, now I can set my mind at ease that I'm not missing out on some nifty radical-simplifying trick. The answer is indeed almost 7--but not quite. Thanks for helping me figure it out.


ReplyThread Parent
laurel_potter
laurel_potter
laurel_potter
Fri, Mar. 27th, 2009 02:06 pm (UTC)

And again, I see there are many smart people on my LJ.

I graduated school in 1971, and only took two math classes -- "math" in 9th grade, in which I got a final grade of "D", then was foolish enough to take algebra in 10th grade and got a comparable final grade.

It's like the other poster above said: My eyes glazed over.


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hymnia
hymnia
Joie
Fri, Mar. 27th, 2009 02:15 pm (UTC)

Well, I hope by calling for "smart people on my LJ" to help with this problem I didn't imply that my friends who aren't great at math aren't smart. I know lots of smart people who don't do so well with algebra and geometry. Also, the time gap does make a huge difference. The main reason math concepts are (usually) easy for me right now is because I've been teaching them for the last few years, so they are very fresh in my mind. When I first started teaching/tutoring after college, I struggled a lot to remember the math, and it had only been four years for me.


ReplyThread Parent
laurel_potter
laurel_potter
laurel_potter
Fri, Mar. 27th, 2009 02:23 pm (UTC)

...I didn't imply that my friends who aren't great at math aren't smart.

Oh, no, not at all. I really love seeing something like this come up -- something I don't even begin to understand -- and see how many people respond.

For the record, I got bad grades in everything in high school, except for typing, home ec and choir. LOL


ReplyThread Parent
hymnia
hymnia
Joie
Fri, Mar. 27th, 2009 07:03 pm (UTC)

Icon <3!

*is music dork*


ReplyThread Parent
lonelychick1917
lonelychick1917
Ally
Sat, Mar. 28th, 2009 03:08 pm (UTC)

lol, I was sitting there with my eyes wide open while you were trying to figure it out, going "ZOMG! She doesn't know the answer!!! PWNED!!"

lol, jk. :D Your an awesome "mathinist". (Will just thought that up, I thought it was good!)

BTW, just in case anyone is reading this comment and going "whaa...?", I'm in her geometry group. Thought I'd clear that up.

^,^


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weverlyware
weverlyware
Tue, Apr. 7th, 2009 01:37 pm (UTC)

Holy crap, I forget all that stuff, but I probably could find out...


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hymnia
hymnia
Joie
Wed, Apr. 8th, 2009 02:54 am (UTC)

It makes sense when you explain it that way, but if that's the case, then I don't understand why the method of simplifying with fractional exponents (in this comment) would come out to 2400 to the power of 1/4, which is not seven. That method didn't involve calculator approximations, so I don't see any reason why it would be off the mark.

I was just speaking about you and mark the other day. Mark hired a close friend of mines little brother, steven mylett. (you might remember Jason)
You should tell Mark to look out for him, he's a good guy.


Yeah, I remember Mark told me about that a while back. I'll mention it to him next time I see him. :)


ReplyThread Parent