Linked Questions

13
votes
8answers
18k views

How do you simplify this square root of sum: $\sqrt{7+4\sqrt3}$?

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 ...
13
votes
5answers
10k views

Simplifying $\sqrt[4]{161-72 \sqrt{5}}$

$$\sqrt[4]{161-72 \sqrt{5}}$$ 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 ...
9
votes
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
59
votes
2answers
3k views

Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by ...
14
votes
4answers
2k views

Simplify $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$.

Denest $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$. 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.
4
votes
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 ...
1
vote
2answers
4k views

How to solve $\sqrt{9-4\sqrt{5}}=$?

Need some hints how to solve this: $\sqrt{9-4\sqrt{5}}=$ ? Thanks.
7
votes
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 ...
7
votes
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 ...
4
votes
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)...
3
votes
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.
4
votes
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 ...
4
votes
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:...
12
votes
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 $$x=\sqrt[8]{2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{\ddots}}}}}$$ And express as $\...
9
votes
1answer
2k views

Denesting Phi, Denesting Cube Roots

I have been looking into denesting square roots but I have found that $\sqrt[3]{2+\sqrt{5}}$ equals $(1+\sqrt{5})/2$. The same is true for $\sqrt[3]{2-\sqrt{5}}$ and $(1-\sqrt{5})/2$. I cannot figure ...

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