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satellite73
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prove \(\sqrt{5\sqrt{21}}=\frac{\sqrt{7}\sqrt{3}}{\sqrt{2}}\)
 2 years ago
 2 years ago
satellite73 Group Title
prove \(\sqrt{5\sqrt{21}}=\frac{\sqrt{7}\sqrt{3}}{\sqrt{2}}\)
 2 years ago
 2 years ago

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satellite73 Group TitleBest ResponseYou've already chosen the best response.1
any help ? i have seen something similar. here is what wofram says http://www.wolframalpha.com/input/?i=sqrt%285sqrt%2821%29%29
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.0
Aren't you asking the same question in different form?
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
for two reasons. it might be easier if we know the answer and "simplify" doesn't mean anything in this context
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.0
Sorry, but one simple question: in reality, is it possible to do it backward?
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
this looks for all the world like one of those trig questions where you use one half angle formula and another half angle formula and you get two different looking but equal answers. then it is a pain to show they are the same, but it usually amounts to multiplying repeatedly
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
ok here is a "well known formula from basic algebra" \[\sqrt{a+b\sqrt{c}}=\sqrt{\frac{1}{2}(a+\sqrt{q})}\sqrt{\frac{1}{2}(a\sqrt{q})}\] if \(q=a^2b^2c\) is a perfect square
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
i must have missed that day
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
guess i have my homework cut out for me
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
typo actually it should be \[\sqrt{a\pm b\sqrt{c}}=\sqrt{\frac{1}{2}(a+\sqrt{q})}\pm\sqrt{\frac{1}{2}(a\sqrt{q})}\]
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
given this formula we are done, since \(5^221=4\) a perfect square, so we get the answer immediately. how we know this i am not sure
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.0
Sorry, I don't want to do so. but.. Please check this http://openstudy.com/study#/updates/4fda8ca0e4b0f2662fd103f2
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
ok lets try this start with \[\sqrt{5\sqrt{21}}\] and write the radicand as a perfect square. so \[\sqrt{5\sqrt{21}}=xy=\sqrt{(xy)^2}=\sqrt{x^2+y^22xy}\] now we can put \(x^2+y^2=5, 2xy=\sqrt{21}, y=\frac{\sqrt{21}}{2x}\) and so \(x^2+(\frac{\sqrt{21}}{2x})^2=5\) and so \[4x^420x^2+21=0\] solving gives \(x=\sqrt{\frac{7}{2}}\) i think
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
i will have to look at this more the first "well known formula" comes directly from this paper, but no explanation, so not much help except in getting the answer
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
the second method i tried to mimic what i read here http://gauravtiwari.org/2011/03/16/aproblemonordinarynestedradicals/ but the line "which on simplification yields..." was not so obvious to me
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
ok, take a look at 4.5.2 here and the following theorem. this i think give it all
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
learn something new every day!
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
today for example i learned where to get a really great hamburger
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
your friend must have run in to someone who was either nuts ( a nutty math teacher, imagine) or just learned this and wanted to show off. i cannot imagine this in a basic algebra class
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.0
*Disappointed* You're not learning HOW to get a great hamburger but you learnt WHERE to get it :( Thanks for all the notes and the 'well unknown formula' which I haven't heard of. Hmm.. Actually, I can solve it :
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
great. let me know how it goes. if they had had the internet when i was in school i would have 4 degrees by now
 2 years ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.0
This might be good enough. Look at \(\sqrt{5\sqrt{21}}\). If you square it, you get \(5\sqrt{21}\). We want this to be a perfect square of some binomial with square roots. This means, that we should want a square multiplied by two. Hence, multiply this by two. We get \[102\sqrt{21}\]If we want this to be a perfect square of some expression that looks like \(\sqrt x\pm\sqrt y\), we want to find an \(x, y\) such that \(x+y=10\) and \(xy=21\). Solving this, we get that \[102\sqrt{21}=(\sqrt7\sqrt3)^2\]Now we work backward to get what we had at the start. Divide by 2, and then take the square root gives us that \[\sqrt{5\sqrt{21}}=\frac{\sqrt7\sqrt3}{\sqrt2}\]
 2 years ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.0
Typo in that first paragraph. I said "This means, that we should want a square multiplied by two." It should read "This means, that we should want a square root multiplied by two." (talking about the square root in \(5\sqrt{21}\) of course)
 2 years ago

Callisto Group TitleBest ResponseYou've already chosen the best response.0
@KingGeorge I solved the question using this way. But no one here looks at it except myininaya
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.1
this looks much snappier
 2 years ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.0
The two methods are very slightly different. Same trick, but slightly different process.
 2 years ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.0
Namely, I took things out of the square root, and you left them in. So I guess they really are just the same thing.
 2 years ago
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