Linked Questions

73 votes
6 answers
22k views

Strategies to denest nested radicals $\sqrt{a+b\sqrt{c}}$

I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$....
JSCB's user avatar
  • 13.6k
14 votes
9 answers
30k 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 ...
BoltKey's user avatar
  • 421
5 votes
6 answers
5k 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 ...
user332176's user avatar
8 votes
6 answers
2k views

How to factorize this $\sqrt{8 - 2\sqrt{7}}$?

When I was at high school, our teacher showed us a technique to simplify square roots like this $\sqrt{8 - 2\sqrt{7}}$ that I forgot. It was something like 8 = 7+1; 7 = 7*1; and using them we could ...
Baimyrza Shamyr's user avatar
4 votes
3 answers
11k 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 ...
MayankJain's user avatar
3 votes
2 answers
13k 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.
Mob's user avatar
  • 375
6 votes
3 answers
588 views

how to simplify $ 2 \sqrt{2}\left(\sqrt{9-\sqrt{77}} \right) $

How to simplify $$ 2 \sqrt{2}\left(\sqrt{9-\sqrt{77}} \right) $$ so that it has no nested radicals? This question is same as that already posted but with a different point of view.
M.Hamza Ali's user avatar
3 votes
2 answers
1k views

Intermediate fields of splitting field

I'm trying to find the intermediate fields of the extension $\mathbb Q\big /\mathbb Q(\alpha)$, where $\alpha = \sqrt{7+\sqrt{13}}$. To do so I've tried to use the Galois correspondence. I've already ...
synack's user avatar
  • 984
7 votes
3 answers
710 views

Square and cubic roots in $\mathbb Q(\sqrt n)$

Here is my question : Let $n$ a squarefree positive integer, $m \ge 2$ an integer and $a+b \sqrt n \in\mathbb Q (\sqrt n).$ What (sufficient or necessary) conditions should $a$ and $b$ satisfy so ...
Watson's user avatar
  • 23.9k
2 votes
2 answers
2k views

Convert from Nested Square Roots to Sum of Square Roots

I am looking for a way to easily discover how to go from a nested root to a sum of roots. For example, $$\sqrt{10-2\sqrt{21}}=\sqrt{3}-\sqrt{7}$$ I know that if I set $\alpha=\sqrt{10-2\sqrt{21}}$,...
Lalaloopsy's user avatar
  • 1,893
0 votes
3 answers
2k views

Find the square root of $14+6\sqrt5 $

Find the square root of $14+6\sqrt5$ I don't just want the answer, can you please tell me the method how you get the answer
user476799's user avatar
-3 votes
4 answers
2k views

Please give me the method of writing $\sqrt{33 + 12 \sqrt 6}$ in the form $a + b \sqrt c$ [closed]

I have the answer but I need the method. My professor wants it in : a + b √c. He said a hint would be to use algebra.
user372636's user avatar
5 votes
1 answer
301 views

General method for writing $\sqrt{a+\sqrt{b}}$ as $c + \sqrt{d}$

Is there a general way to write $\sqrt{a+\sqrt{b}}$ as $c + \sqrt{d}$ for $a,b,c,d$ positive integers? Is it always possible? I've seen several ways to solve specific cases like $$\sqrt{6+4\sqrt{2}} = ...
omega-stable's user avatar
  • 1,235
0 votes
2 answers
310 views

Finding the square root of $a \pm b\sqrt c$. [closed]

I have an exercise that says : Simplify $\sqrt{16+2\sqrt{55}}$. Please, I need a vivid explanation. Can anyone help?
user9602923's user avatar
3 votes
3 answers
75 views

Given area of square $= 9+6\sqrt{2}$ Without calculator show its length in form of $(\sqrt{ c}+\sqrt{ d})$

$\sqrt{9+6\sqrt{2}}$ to find length But how do I express the above in the form of $\sqrt{c} + \sqrt{d}$.
Jayle's user avatar
  • 41

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