Questions tagged [nested-radicals]

In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression.

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Nested radical question (1993 All-Russian Olympiad)

I received this interesting problem in an email. It comes from the $1993$ All-Russian Olympiad grade 10, round 4. Prove that: $$\sqrt{2 + \sqrt[3]{3 + \cdots + \sqrt[1993]{1993}}} < 2$$ I did ...
Presh's user avatar
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Any tips to solve this algebra problem? [duplicate]

How to solve this? I have tried making them in the form $(a+b)^3$ and $(a-b)^3$ but have been unsuccessful so far. Any help will be deeply appreciated. Thanks! $$ \sqrt[3]{5\sqrt{2}+7}-\sqrt[3]{5\sqrt{...
Nikhil Panigrahy's user avatar
2 votes
0 answers
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Find all integer values $x$, such that $\sqrt{x + \sqrt{x+ \ldots+\sqrt{x}}} < n$, for any number of radicals

Consider an integer ${n}\ge{2}$. Find all integers $x$ such that, for any number of radicals: $$\sqrt{x + \sqrt{x+ \ldots+\sqrt{x}}}<n$$ I know that this problem is supposed to be solved using ...
MushroomTea's user avatar
2 votes
4 answers
161 views

Find the result of$\sqrt[3]{77-20\sqrt{13}}+\sqrt[3]{77+20\sqrt{13}}$ [duplicate]

How to find the result of$$\sqrt[3]{77-20\sqrt{13}}+\sqrt[3]{77+20\sqrt{13}}$$ I tried using $$\begin{align}a\pm b\sqrt{13}&=\sqrt[3]{(a\pm b\sqrt{13})^3}\\&=\sqrt[3]{(a^3+39ab^2)\pm \sqrt{13}(...
Namura's user avatar
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Does this ridiculous integral converge?

I was looking through some of my older questions, when I came across this crazy integral I posted. $\int\limits^{2}_{1}\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}dx$ I had approximated it at ...
Dylan Levine's user avatar
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1 vote
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$h^2=(2\sqrt[3]{2})^2 - (\sqrt[3]{2})^2$. h should be equal to $\sqrt[6]{108}$ but I get $\sqrt[6]{4}$ as a result. Please help.

My resolution $${\quad \quad \quad(2\sqrt[3]{2})^2 = h^2+(\sqrt[3]{2})^2 \\ \iff ( \sqrt[3]{16})^2= h^2 + \sqrt[3]{4} \\ \iff \sqrt[3]{256} = h^2 + \sqrt[3]{4}\\ \iff h^2 = \sqrt[3]{256}- \sqrt[3]{4}\...
Manuel's user avatar
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Is there any existing definition for unique simplest nested radical expression?

From a wikipedia entry section about the simplified form of a radical expression, only non-nested radical expression is covered. There are no hints about nested radical expressions. Some common sense ...
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3 votes
2 answers
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Getting rid of cube roots in the form of (a+b)+(a-b)

So, I've come across a question that asked "Simplify the sum $\sqrt[3]{18+5\sqrt{13}}+\sqrt[3]{18-5\sqrt{13}}$". As I've had little to no experience with these kinds of questions, I would ...
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Prove that $2\sin\left(\frac{\pi}{4}x\right)$ can be expanded as nested radical, where the order of $+,-$ can be described in certain way

I am asked to prove that $$ 2\sin\left(\frac{\pi}{4}x\right)=\lim_{n\to\infty}\bigg(s_0\sqrt{2+s_1\sqrt{2+s_2\sqrt{2+\cdots+s_{n-1}\sqrt{2+s_n}}}}\bigg) $$ for $x\in[-2,2]$, where $\{s_n\}_{n\ge0}\...
SlInevitable2003's user avatar
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How do we prove this nested radical solution?

I’m submitting this question because my previous one was perhaps too verbose and did not get the point very quickly. A well known $\sqrt{2}$ nested radical has the following solutions as found in the ...
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A Nested Radical Arising from a Nonlinear Recurrence

I have been looking into analytic continuation of regular recurrence relations into the negative, real, and complex domains ($\mathbb{N} \mapsto \mathbb{Z}, \mathbb{R},\mathbb{C}$). In doing so, we’ve ...
Cye Waldman's user avatar
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6 votes
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Find the fourth roots of the following binomial surd: $14+8\sqrt{3}$

Find the fourth roots of the following binomial surd: $X=14+8\sqrt{3}$ I attempt to find the square root first: $\sqrt{X}=\sqrt{14+8\sqrt{3}}$ $\sqrt{14+8\sqrt{3}}=\sqrt{x_1}+\sqrt{y_1}$ $(\sqrt{14+8\...
ronald christenkkson's user avatar
4 votes
2 answers
279 views

Show $\frac{\pi^2}{6}\sqrt{ab+\sqrt{ab+2\sqrt{ab+3\sqrt{\cdots}}}}>4$ by hand .

Problem : Show that if : $$a=\int_{0}^{1}x!dx,b=\int_{0}^{\infty}1/\Gamma(x)dx,x!=\Gamma(x+1)$$ Then we have : $$S=\frac{\pi^2}{6}\sqrt{ab+\sqrt{ab+2\sqrt{ab+3\sqrt{\cdots}}}}=4.0054\cdots>4$$ ...
DesmosTutu's user avatar
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2 answers
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Calculating $\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{100\;\text{times}}$ without a calculator?

Is there any way to find the summation up to $100$ times without using calculator? $$\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{100\;\text{times}}$$ I know that if the above summation ...
DEEPANWITA Roy's user avatar
6 votes
4 answers
200 views

How does $\sqrt[6]{26+15\sqrt3}-\sqrt[6]{26-15\sqrt3}$ become $\sqrt2$?

How can I simplify this? $$\sqrt[6]{26+15\sqrt3}-\sqrt[6]{26-15\sqrt3}$$ I know the answer is $\sqrt2$, but don't know where to start. Thx.
Gerard's user avatar
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Mixing Continued fraction and nested radicals does $D=-\pi+\ln(16)+(7\ln\pi)/2-7/3\arctan(\pi)$?

Let : $$B=\cdot\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\cdots}}}}}}}}$$ Let : $$D=\frac{1}{B+\frac{1}{B+\frac{1}{B+\frac{1}{B+\cdots}}}}$$ Then I conjecture with WA that ...
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2 votes
2 answers
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Can the nested radicals in this expression be simplified?

I have the following expression: $$ \sqrt{ 1 X^2 - \sqrt[3]{X^2 Y + \sqrt{\left(X^2 Y\right)^2 + \left(\frac{2}{3} Y\right)^3}} - \sqrt[3]{X^2 Y - \sqrt{\left(X^2 Y\right)^2 + \left(\frac{2}{3}...
Lawton's user avatar
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3 votes
5 answers
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How to find the maximum value of $x^2-\frac {1}{x^2}$ if the real number $x$ satisfies $x^3-\frac{1}{x^3}=\frac {28}{27}$

Suppose the real number $x$ satisfies $x^3-\dfrac{1}{x^3}=\dfrac {28}{27}.$ Find the maximum value of $x^2-\dfrac {1}{x^2}.$ A) $\frac{1}{6}\sqrt{37}$ B) $\frac{1}{9}\sqrt{37}$ C) $\frac{1}{3}\sqrt{...
mathtime's user avatar
1 vote
0 answers
129 views

A very interesting integral: $\int\limits^{2}_{1}\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}dx$

I really like infinitely nested radicals so when I was messing around with them in Desmos, I noticed that the domain of $\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}$ seemed to converge to $1&...
Dylan Levine's user avatar
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1 answer
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Nested radicals $\sqrt{C+1+b\sqrt{C+1+b^{2}\sqrt{C+1+b^{3}\sqrt{\cdot\cdot\cdot}}}}=g(x)$ then $\lim_{x\to\infty}(g(x+1)-g(x))=^?1$

Well let the problem first : Conjecture : Let $x>M>0$, $b=\sqrt{x}$,$0<C<1$ then define : $$\sqrt{C+1+b\sqrt{C+1+b^{2}\sqrt{C+1+b^{3}\sqrt{\cdot\cdot\cdot}}}}=g(x)$$ Then it seems we have :...
DesmosTutu's user avatar
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9 votes
10 answers
803 views

What is the simple way to solve the equation $x+\sqrt{a+\sqrt{x}}=a$ for all real $x$?

Solve the equation $$x+\sqrt{a+\sqrt{x}}=a$$ for all real $x$ and nonzero parameter $a$, prove that the equation has a solution if and only if $a\ge 1$. My attempts and thoughts. $$\sqrt {a+\sqrt x}=...
user1178051's user avatar
1 vote
4 answers
154 views

Find the square root of the following binomial surd: $\sqrt{27}+2\sqrt{6}$

Find the square root of the following binomial surd: $\sqrt{27}+2\sqrt{6}$ $\sqrt{\sqrt{27}+2\sqrt{6}}=\sqrt{x}+\sqrt{y}$ $(\sqrt{\sqrt{27}+2\sqrt{6}})^2=(\sqrt{x}+\sqrt{y})^2$ $3\sqrt{3}+2\sqrt{6}=x+...
ronald christenkkson's user avatar
0 votes
0 answers
102 views

solutions to $x = \sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}\;\;(\text{mod}\;p)$

Fix an integer $0<a<p-1$, and an odd prime $p$. Define $$S(a,p)=\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}\;\;(\text{mod}\;p)$$ to be the set of all integers $x\in\{0,...,p-1\}$ such that, for some ...
mick's user avatar
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7 votes
1 answer
220 views

Piecewise function with $\sqrt{2 + \sqrt{n}}$ and $\sqrt{1 + \frac{\sqrt{n}}{2}} + i \sqrt{-1 + \frac{\sqrt{n}}{2}}$

Problem Let $f : \mathbb{C} \setminus \{0\} \rightarrow \mathbb{C}$, $f(z) = z + \frac{1}{z}$. Given that $$ f(z) = \begin{cases} \sqrt{2 + \sqrt{n}} & \text{if } z \in \mathbb{R} \setminus \...
JHumpdos's user avatar
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-2 votes
1 answer
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Nested Square Roots

Question: Solve for $\sqrt{x\sqrt{x\sqrt{x}}}$ I know this may be a duplicate but I have not found one yet. Additionally I understand the logic for base 8 but why would the answer be $x^\frac 78$ ...
PleaseSir MayIHaveSomeMore's user avatar
1 vote
2 answers
205 views

limit of the sequence $a_n = \sqrt{2+\sqrt{3+\sqrt{2+\sqrt{3+...}}}}$

I have already shown that the sequence is monotone and bounded, so it is convergent. what I need is to calculate the limit when $a_n = \sqrt{2+\sqrt{3+\sqrt{2+\sqrt{3+...}}}}$, $n \rightarrow \infty$, ...
JAISON ALEXANDER MUNOZ HORMIGA's user avatar
2 votes
0 answers
35 views

What is the rate of convergence of the following sequence (equation with a finite number of nested radicals)?

Let $f(x)=\sqrt{1-x^2}$, $b = 1/\sqrt{2}$. The sequence $(E_n)_{n=1}^{\infty}$ is defined as the solution to the following equation : $$f(E_n - f(E_n -f(E_n - ....-f(E_n - b)))) = E_n -1,$$ where the ...
Marc_Adrien's user avatar
1 vote
1 answer
101 views

Can this set of equations be extrapolated to a complete pattern?

Quick Background We have five independent variables that can each be any real number greater than zero: $d_{max}$ $v_{max}$ $a_{max}$ $j_{max}$ $s_{max}$ These variables are linked to a chain of ...
Lawton's user avatar
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1 vote
4 answers
121 views

How to simplify the cubic radical $\sqrt[3]{a\sqrt{b}-c}$?

How to simplify the cubic radical $\sqrt[3]{a\sqrt{b}-c}$ ? So I encountered a particular problem in chapter of surds and radicals to find the cube root of $38\sqrt{14}-100\sqrt{2}$ . So I took out 2√...
Ash_Blanc's user avatar
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-1 votes
1 answer
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An Olympiad Question on cube roots. [closed]

If $$\sqrt[3]{\sqrt[3]{2}-1} = \sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{\frac{1}{9}}$$where $a$ is greater than $b$, then find the value of $a+2b+3$. I got this question on an olympiad handout and couldn't ...
Pulkit Sabharwal's user avatar
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0 answers
52 views

An Infinite Sum of Nested Radicals (With A Partial Solution)

Define $f(y) = \sqrt{y^{2}+\frac{2x}{N}+-\frac{2yx}{N}}$, where $x$ is any positive real number and $N$ is the number of times we wish to compose $f$ with itself. I wish to find the limit as the ...
Definitely Not a English Major's user avatar
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0 answers
63 views

Equality with nested cubic radicals. Where is my error?

We have $$\sqrt[3]{ \frac{1 - 3 \sqrt[3]{3}}{11- 3 \sqrt[3]{3}}} = \frac{1 - \sqrt[3]{3}}{2- \sqrt[3]{3}}$$ since $$\require{cancel} \sqrt[3]{ \frac{\ 1 - 3^{1/3} 3}{11 - 3^{1/3} 3} }=\frac{(\ 1 - 3^...
orangeskid's user avatar
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1 vote
1 answer
70 views

Computationally representing a Fuchsian group

I'd like to learn the math behind this code golf answer, in which the symmetry group of the order-4 pentagonal tiling is represented by integer matrices. My understanding so far is: In $PSL(2,\Bbb R)$ ...
Karl's user avatar
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Closed form for an alternating values infinite sequence of nested roots

I tried to derive a closed form for an infinite sequence of nested square roots with alternating values and found myself with a 4th degree equation, which I'm fine with, but I was wondering if there's ...
AndreaC1979's user avatar
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0 answers
71 views

Ramanujan's nested radical: complex numbers this time

It is well-established that when the square root defined on positive real numbers outputs positive real numbers, the equation below holds: $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$ for a ...
oneequalstwo's user avatar
5 votes
1 answer
131 views

Nested Radicals from Brazil

Show that $$ \frac{ \sqrt{2} }{ \sqrt{\sqrt[4]{ \frac{\sqrt{5}+2}{4}} + 1} - \sqrt{\sqrt[4]{ \frac{\sqrt{5}+2}{4}} - 1}} = \sqrt[8]{ 1 + 2 \sqrt{ \sqrt{5} -2 } }. $$ What I've tried so far. $$ \begin{...
QED's user avatar
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3 votes
1 answer
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Sums with nested radicals related to $1/\pi^2$

Context: I evaluated some sums related to $1/\pi^{2}$ that for a reason I don't know I can't find in the literature. $$(1)\hspace{.5cm}{1\over 2\sqrt{2}}+{1\over 2^{3}\sqrt{2+\sqrt{2}}}+{1\over 2^{5}\...
user avatar
4 votes
0 answers
53 views

Algebraic methods for evaluating infinite nested square root radicals

Intrigued by an challenge in the Dutch Math Olympiade, I studied a way to evaluate infinite nested radicals. The question was to evaluate $\sqrt{4+\sqrt{18+\sqrt{40+\sqrt{70+\sqrt{108+\sqrt{156+…...}}}...
Paul vdVeen's user avatar
8 votes
1 answer
300 views

Nested radical question from Harvard MIT maths competition

The question is $ f(n) = \sqrt{{100}+\sqrt{n}} + \sqrt{{100}-\sqrt{n}} $ What is the minimum value of n for which $f(n)$ is an integer. $n$ is a natural number. I made an attempt to solve it, and ...
Chris Daniel's user avatar
0 votes
0 answers
79 views

Geometric series summation with power which is also geometric.

I was just pondering how will this series converge. Assuming $n \ge 0$, find $$ f(n)=n^{1/2} + n^{1/4} + n^{1/8} + n^{1/16} +\text{...}+ 1 $$ We stop the summation when we hit $1$ i.e. $\lfloor{n^{1/...
Ukh's user avatar
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6 votes
1 answer
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Is this general nested radical for $\pi$ true?

We have, I. Liu Hui (c. 300 AD) $$\pi \approx 3\cdot2^{\color{red}8}\times \underbrace{\sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2+\sqrt{\color{blue}1}}}}}}}...
Tito Piezas III's user avatar
3 votes
1 answer
53 views

Effective degree bound for solvability by radicals

Let $P\in{\mathbb Q}[X]$ be an irreducible polynomial of degree $n\geq 3$, and let $\mathbb L$ be the decomposition field of $P$. Denote the Galois group of the extension ${\mathbb L}:{\mathbb Q}$ by $...
Ewan Delanoy's user avatar
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0 votes
1 answer
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Coefficient cubic for the nested radical equation $\pm \sqrt{n\pm \sqrt{n\pm\sqrt{n\pm \sqrt{n+x}} }}=x$

I read a math article on the net that stated: For the nested radical equation $$\pm \sqrt{n\pm \sqrt{n\pm\sqrt{n\pm \sqrt{n+x}} }}=x$$ By repeated squaring we get $$\left( \left( \left( x^2-n \right)^...
Nanhui Lee's user avatar
1 vote
1 answer
178 views

How do we know if a system of equations has multiple solutions and how to solve it?

For example, my equations are $\lambda_1=\sqrt{\dfrac{\alpha^2}{2}+\sqrt{k^4+\dfrac{\alpha^4}{4}}}$ $\lambda_2=\sqrt{-\dfrac{\alpha^2}{2}+\sqrt{k^4+\dfrac{\alpha^4}{4}}}$ $\lambda_1\tan\!\big(\!\...
Gopalpur's user avatar
5 votes
1 answer
153 views

Proving Rate of Convergence

I am investigating the following coupled sequence: \begin{align*} y_0 &= 1\\ x_{n+1} &= \sqrt{1 + \frac{1}{y_n}}\\ y_{n+1} &= \sqrt{1 - \frac{1}{x_{n+1}}}\\ \end{align*} I am trying to ...
Sharky Kesa's user avatar
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0 answers
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Is there a name for a radical field extension of the rationals that contains all radical/solvable elements?

I am wondering if there exists a name for the field $F$ such that $\mathbb{Q}\subset F\subset \mathbb{A}$, and $F$ contains all the radical elements such as $\sqrt[7]{2}, \sqrt[3]{3-\sqrt[4]{7}}, \...
Leon Kim's user avatar
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3 votes
1 answer
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Integral of $\displaystyle \int\sqrt{x\sqrt{x\sqrt{x...\sqrt{x}}}}\,dx$ where $x$ is repeated $n$ times

Compute the indefinite integral $$\displaystyle \int\sqrt{x\sqrt{x\sqrt{x\cdots\sqrt{x}}}}~~dx=\displaystyle \int\sqrt{x\underset{n~\text{times}}{\underbrace{\sqrt{x\sqrt{x\sqrt{x...\sqrt{x}}}}}}}~~dx$...
Bouzari Abdelkader's user avatar
1 vote
0 answers
77 views

Is there any general way of finding root of a irrational number? (eg, getting $\sqrt2+\sqrt3$ from $\sqrt{5+2\sqrt6}$) [duplicate]

for example: $(\sqrt2+\sqrt3)^2 = 5+2\sqrt6$ is straightforward. But how can we get $\sqrt2+\sqrt3$ from $\sqrt{5+2\sqrt6}$? Is there any general method to do it?
ICFSZ's user avatar
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5 votes
8 answers
245 views

How does $\sqrt{2+\sqrt{2}}+\sqrt{2-\sqrt{2}}$ become $\sqrt{2(2+\sqrt{2})}$?

I'd like to know how can one simplify the following expression $$\sqrt{2+\sqrt{2}}+\sqrt{2-\sqrt{2}}$$ into $$\sqrt{2(2+\sqrt{2})}.$$ Wolfram alpha suggests it as an alternative form, and numerically ...
coder's user avatar
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5 votes
3 answers
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Finding the value of $\sqrt[4]{(4+\sqrt7)^{-1}}\sqrt{1+\sqrt7}$ with other approaches

It is a problem from a timed exam, What is the value of $\sqrt[4]{(4+\sqrt7)^{-1}}\sqrt{1+\sqrt7}$ ? $1)1\qquad\qquad2)\sqrt[4]2\qquad\qquad3)2\qquad\qquad4)2\sqrt[4]2$ I solved it with two ...
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