# Linked Questions

34 questions linked to/from Strategies to denest nested radicals.
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I came around this expression when solving a problem. $$\sqrt{7+4\sqrt{3}}$$ WolframAlpha says it equals $2+\sqrt{3}$. We can confirm it like this $$\left(2+\sqrt{3}\right)^2 \;=\; 4+4\sqrt{3} + 3 ... 5answers 10k views ### Simplifying \sqrt{161-72 \sqrt{5}}$$\sqrt{161-72 \sqrt{5}}$$I tried to solve this as follows: the resultant will be in the form of a+b\sqrt{5} since 5 is a prime and has no other factors other than 1 and itself. Taking this ... 7answers 2k views ### How to simplify a square root How can the following:$$ \sqrt{27-10\sqrt{2}} $$Be simplified to:$$ 5 - \sqrt{2} $$Thanks 2answers 3k views ### Denesting radicals like \sqrt{\sqrt{2} - 1} The following result discussed by Ramanujan is very famous:$$\sqrt{\sqrt{2} - 1} = \sqrt{\frac{1}{9}} - \sqrt{\frac{2}{9}} + \sqrt{\frac{4}{9}}\tag {1}$$and can be easily proved by ... 4answers 2k views ### Simplify \sqrt {\sqrt{5}-\sqrt{4}}. Denest \sqrt {\sqrt{5}-\sqrt{4}}. I have tried completing square by several method but all failed. Can anyone help me please? Thank you. p.s. I'm a poor question-tagger. 6answers 2k views ### Simplifying radicals inside radicals: \sqrt{24+8\sqrt{5}} Simplify: \sqrt{24+8\sqrt{5}} I removed the common factor 4 out of the square root to obtain 2\sqrt{6+2\sqrt{5}}, but the answer key says it is 2+2\sqrt{5}. Am I missing out on some general rule ... 2answers 4k views ### How to solve \sqrt{9-4\sqrt{5}}=? Need some hints how to solve this: \sqrt{9-4\sqrt{5}}= ? Thanks. 2answers 5k views ### Value of \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}} Here is the question:$$\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}} = \;?$$(original image) I think ... 2answers 3k views ### Denesting a square root: \sqrt{7 + \sqrt{14}} Write:$$\sqrt{7 + \sqrt{14}} = a + b\sqrt{c}$$Form.$$7 + \sqrt{14} = a^2 + 2ab\sqrt{c} + b^2c$$a^2 + b^2c = 7 and 2ab = 1, and c = 14 But that doesn't seem right as a, b, wont be ... 6answers 241 views ### How to prove 2\sqrt{2+\sqrt{3}}=\sqrt{2}+\sqrt{6}? My calculator and I were arguing one day about the cosine of some number. The calculator said "\cos(\frac x2)=\sqrt{2}+\sqrt{6}". I said "That's absurd because \cos(\frac x2)=\sqrt{\frac{1+\cos(x)... 2answers 11k views ### Square root of surds: \sqrt{12+2\sqrt{6}}? [closed] I got this question Find the square root of 12+2\sqrt{6} expressing your answer in the form \sqrt{m}+\sqrt{n}. I have no idea what this means and how to go about it. 3answers 6k views ### How to evaluate \sqrt{5+2\sqrt{6}} + \sqrt{8-2\sqrt{15}}? My exams are approaching fast and I found this question in one of the unsolved sample papers. I tried squaring the whole term but couldn't work out the answer. I am a ninth grader so please try to ... 6answers 417 views ### How do I show that \sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2} According to wolfram alpha this is true: \sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2} But how do you show this? I know of no rules that works with addition inside square roots. I noticed I could do this:... 2answers 602 views ### Continued fraction in 8th root---— any simpler approach? It seems to be a problem from the Putnam Exam. The problem asked to find the exact value of$$x=\sqrt{2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{\ddots}}}}} And express as $\... 1answer 2k views ### Denesting Phi, Denesting Cube Roots I have been looking into denesting square roots but I have found that$\sqrt{2+\sqrt{5}}$equals$(1+\sqrt{5})/2$. The same is true for$\sqrt{2-\sqrt{5}}$and$(1-\sqrt{5})/2\$. I cannot figure ...

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