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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Seeking confirmation my answer is correct and well-formatted

The following is a question that I composed and solved. I want to know if it is mathematically correct and well-formatted. Question: Show that $\root{3}\of{\sqrt{5}+2}+\root{3}\of{\sqrt{5}-2}=\sqrt{5}$...
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Generalizing Ramanujan cubic denesting formula to higher powers

We have the following theorems for denesting radicals of degree 2 and 3 : Denesting theorem for degree 2 : If $\alpha, \beta$ are the roots of the equation, \begin{equation} x^2-ax+b = 0 \end{equation}...
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A new (?) infinitely nested radical equals $1$

Let $x$ be a real number such that $x\ge{0}$, then $$1=\sqrt{\frac{\sqrt{\frac{\sqrt{\frac{\sqrt{\frac{x+1}{2}}+1}{2}}+1}{2}}+1}{2}}+...$$ At least I haven't seen it on the internet. Questions: a) Is ...
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General formula for the upper bound of pi involving nested square roots (circumscribed perimeters of regular polygons)

The formula for the lower bound of pi involving nested square roots looks like this: $p_{2^m} = 2^m\sqrt{2-\sqrt{2+\sqrt{2+ \sqrt{2+...}}}}$ where there are $m-1$ nested square roots. For example, ...
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18 votes
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A Proof with no words that $\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=2$

Question What are the words to describe the method in the image below? (from Nelsen's Proofs without Words II) Attempt I was thinking and could define the sequence $u_1=2; u_{n+1}=f\circ g^{−1}(u_n)$ ...
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4 votes
3 answers
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Solving $3 + \sqrt{3^2 + \sqrt{3^4 + \sqrt{3^8 + \sqrt{3^{16} + ...}}}}$

How to find the value of $3 + \sqrt{3^2 + \sqrt{3^4 + \sqrt{3^8 + \sqrt{3^{16} + ...}}}}$ I tried to solve it and found a relation that if I assume the given expression to be something say $x$, then ...
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Looking for 2 nested radicals neither of which denest but their sum DOES denest.

By nested radical, I mean an expression of the form $\sqrt{a+b\sqrt{n}}$ where a, b and n are positive integers and n is not a perfect square. I wrote a computer program that randomly generated pairs ...
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How does $ f(x)=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}. x \ge1 $ produce a piecewise function with an interval that is a constant fucntion?

I found this function in an old math contest. $$ f(x)=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}},\space \space x\ge 1 $$ This function is identical to the piecewise function, $$ g(x) = \begin{cases} 2, ...
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Convergence of Sequence of functions with nested radicals.

I am reading Lascota and Mackey's "Chaos, Fractals, and Noise" (which is great). In an early chapter, they define the following operator (really this is a special case of the Frobenius-...
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Recursive sequence relationship with repeating nested radicals

Here I'd like to introduce: $6-\sqrt{6-\sqrt{6-\sqrt{6-\sqrt{6}}}}$ If continuing the pattern indefinitely, the limit is $4$. The values are successively higher and lower than $4$. In a recursive ...
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Is there a general form for infinite nested radicals of degreen n?

Say we have the infinite nested radical $$\sqrt[n]{a + \sqrt[n]{a + \sqrt[n]{a + \cdots}}}$$ When $n = 2$, this evaluates to $\frac{1 + \sqrt{4a + 1}} {2}$, which is the positive root of the equation $...
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1 vote
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Nested sine and cosine half angle formulas [closed]

I am having trouble with this problem. I know that it is related to the sine and cosine half-angle formulas. I substituted $\frac{\sqrt{3}}{2}$ with $\cos(\frac{\pi}{6})$ and got $\frac{\pi}{24}$ for ...
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1 answer
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Finding a root of a degree 5 polynomial

There are positive integers $m$ and $n$ such that $m^2 −n = 32$ and $$\sqrt[5]{m+\sqrt{n}} + \sqrt[5]{m-\sqrt{n}}$$ is a real root of the polynomial $$x^5 −10x^3 + 20x −40$$ Find $m+n$. Okay I write ...
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What is the result of $\sqrt{-\frac{1}{4}+\sqrt{-\frac{1}{4}+\sqrt{-\frac{1}{4}+\sqrt{-\frac{1}{4}+...}}}}$

Given the golden ratio is equal to $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}$ you get $f(x) = \sqrt{1+f(x)}$. Solving for $f(x)$: $f(x)^2 = x + f(x)$ $f(x)^2 - f(x) = x$ $f(x)^2 - f(x) + \frac{1}{4} = ...
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A nested radical

Find $$\large \lim_{n \to \infty}\sqrt[3]{a_0+\sqrt[3]{a_1+\cdots+\sqrt[3]{a_n}}},$$ where $\large a_n=6(3^{3^n}+1)^2.$ According to Herschfeld's Convergence Theorem, the limit do exists , since $\...
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1 vote
4 answers
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Showing that $\sqrt{2+\sqrt3}=\frac{\sqrt2+\sqrt6}{2}$ [duplicate]

So I was playing around on the calculator, and it turns out that $$\sqrt{2+\sqrt3}=\frac{\sqrt2+\sqrt6}{2}$$ Does anyone know a process to derive this?
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Truncation error for $ S = \sum_{m=0}^\infty \frac{\beta^m}{m!}a^{-\frac{m+1}{k}}\Gamma\left(\frac{m+1}{k}\right)$

Define the infinite sums $S_1$ and $S_2$ as $$ S_1 = \sum_{m=0}^\infty \frac{\beta^m}{m!}a^{-\frac{m+1}{k}}\Gamma\left(\frac{m+1}{k}\right)$$ and $$S_2 = \sum_{m=0}^\infty (-1)^m \frac{\beta^m}{m!}a^{-...
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2 answers
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polynomial nesting technique for $f(x)=\sqrt{x^2+1}-x$

$f(x)=\sqrt{x^2+1}-x$ $x=10,10^2,...,10^6$ I want to calculate $f(x)$ and $\frac{1}{f(x)}$ and I want to use polynomial nesting technique that closest approximation to the real value. I'm beginner in ...
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4 answers
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Solve for $x$ in $\sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x$

Solve for $x$: $$\sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x$$ I tried to substitute $y=x+2$ and then I try to solve the equation by again and again squaring. Then I got equation, $$(y-2)(3y^{14}-(y-...
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nested radical simplification

I'm stuck on this radical simplification $$ (\sqrt{5}+1) \sqrt{10+2\sqrt{5}}$$ In my textbook this product is equal to the $tan(\frac{2\pi}{5})$ and the value is : $$ 4 \sqrt{5+2 \sqrt{5}} $$ I tried ...
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1 vote
1 answer
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Interrelationship between cyclic infinite nested square roots of 2 and Summation of Geometric progression (partial and infinite)

Disclaimer: - As this is a lengthy post I am going to follow Q & A pattern Finite and infinite nested square roots of 2 can be solved to some cosine values as discussed earlier here and here It ...
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1 vote
1 answer
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Suppose $a_1=1$ and $a_n=\sqrt{n-a_{n-1}}$ $\forall n \in\mathbb{Z_{\ge 2}}$. What is the least integer $n$ for which $a_n>4$? ⌊$a_{100}$⌋?

Suppose $a_1=1$ and $a_n=\sqrt{n-a_{n-1}}$ $\forall n \in\mathbb{Z_{\ge 2}}$. What is the least integer $n$ for which $a_n>4$? What is $\left \lfloor a_{100} \right \rfloor$, where $\left \lfloor u ...
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4 votes
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Infinite nested radicals from Putnam exam 1953

Prove that the following sequence is convergent and find the limit: $$\sqrt{7}, \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7+\sqrt{7}}}, ... $$ with $x_{n+2}=\sqrt{7-\sqrt{7+x_{n}}}$. Notice that $$2=\sqrt{7-3}...
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What is the value of the infinite nested radical $\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{\cdots}}}}$? [duplicate]

The value is usually taken to be the limit of the partial sums as the number of terms increases beyond limit. In this case each partial sum is trivially zero, so the value of the infinite nested ...
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2 votes
2 answers
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simplify radicals

In the book $A=B$ (link) page 11 example 1.5.1 it is written By setting $\sin a = x$ and $\sin b = y$, we see that the identity $\sin(a + b) = \sin a \cos b + \sin b \cos a$ is equivalent to $$ \...
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3 votes
1 answer
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$\sqrt[3]{A+B\sqrt{C}}$ generalized denesting formula

Q: What is the generalized formula for denesting $\sqrt[3]{A+B\sqrt{C}}$? Recently I've posted a question about nested radicals in solving the cubic equation. I received a comment about an ingenious ...
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2 votes
2 answers
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does this nested radicle converge? $1+\sqrt{1+\sqrt{\dots}+\sqrt{1+\sqrt{\dots}}}+\sqrt{1+\sqrt{1+\sqrt{\dots}+\sqrt{1+\sqrt{\dots}}}}$

Let $a_0=1$ and $a_n=1+\sqrt{a_{n-1}}+\sqrt{1+\sqrt{a_{n-1}}}$ I want to know if the limit of $a_n$ as n goes to infinity. $$1+\sqrt{1+\sqrt{1+\sqrt{\dots}+\sqrt{1+\sqrt{\dots}}}+\sqrt{1+\sqrt{1+\sqrt{...
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3 votes
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Ramanujan's Nested radical Inequality

Recently I was studying functional equation, in which I came across this paragraph about Ramanujan's Nested Radical in Christopher G. Small's book [Functional Equations and How to Solve Them][1]. $$ \...
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6 votes
1 answer
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When does $r + \sqrt s$ with $r,s \in \mathbb Q$ have a cubic root of the form $u \pm \sqrt v$ with $u,v \in \mathbb Q$?

There are quite a number of related questions in this forum, for example Denesting Cardano's Formula Prove that $\sqrt[3]{45+29\sqrt{2}} +\sqrt[3]{45-29\sqrt{2}} $ is rational. Concering square ...
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2 votes
1 answer
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Denesting Cardano's Formula

For a depressed cubic equation $x^3 + px + q =0$ having exactly one real root, Cardano's formula gives the real root as $$\sqrt[3]{-\frac{q}{2} +\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} + \sqrt[3]{-\...
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1 vote
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Complex number : mth root of nth root

If one defines the $n^{th}$ root of a complex number ($n$ a natural number) so that it coincides with the usual one for the positive real : its imaginary part is $\ge 0$ its real part is the largest ...
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1 vote
1 answer
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$\sqrt{2\sqrt{3\sqrt{4.....\infty}}}$

How can i find the value of $\sqrt{2\sqrt{3\sqrt{4.....\infty}}}$? I had problem in this question because this is not like the question $\sqrt{2\sqrt{2\sqrt{2...\infty}}}$ where the numbers are ...
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3 votes
1 answer
216 views

A hypergeometric beast: Simple closed-form for combination of $_4F_3(1)$?

We have $$ \frac{1}{4} -\frac{1}{4}{_4F}_3\left({-\frac{1}{5},\frac{1}{5},\frac{2}{5},\frac{3}{5}\atop\frac{1}{4},\frac{1}{2},\frac{3}{4}};1\right) -\frac{1}{\sqrt[4] {5}}{_4F}_3\left({\frac{1}{20},...
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1 vote
0 answers
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Show that: $A_1A_2=R\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2+...+\sqrt{2}}}}}$ , the number of square roots being $n–1$.

Let $A_1A_2…A_k$ a polygone inscribed in a circle of radius $R$, with $k=2^n$ . Show that: $A_1A_2=R\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2+...+\sqrt{2}}}}}$ , the number of square roots being $n–1$. This ...
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2 votes
2 answers
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Does the $0$ solution for $\sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x...}}}}}$ hold any meaning?

The function $f(x)=\sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x...}}}}}\qquad$ quickly approaches $f(x) =x$. $$f(x)=\sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x...}}}}}\\=\sqrt{xf(x)}\\ \left[f(x)\right]^2=xf(x)\\\left[...
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1 vote
1 answer
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Please help me clear confusion over principal roots and identities for n-th radicals

From my old high school math textbook: If ${a{\geq }0}$ and $n\in \mathbb{N} ^{\ast }$, then ${\sqrt[{n}] {a}}$ is the non-negative solution of ${{x}^{n}}=a$. It then goes on to infer a number of ...
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4 votes
1 answer
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A question about limit [closed]

Can anyone help this questions? Find the limit of $$1+\sqrt{2+\sqrt[3]{3+\sqrt[4]{4+\sqrt[5]{5+....+\sqrt[n]{n}}}}}$$ I can only solve that this formula is less than 3 but can not find the exact ...
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15 votes
4 answers
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Evaluating $\sqrt{1+\sqrt{2 - \sqrt{3 + \sqrt{4 - \cdots}}}}$

$$x =\sqrt{1+\sqrt{2 - \sqrt{3 + \sqrt{4 - \sqrt{5+ \sqrt{6 - \cdots}}}}}}$$ Find $x$. I am not sure how to proceed. Is this a sort of Arithmetico-Geometric Progression? Will this converge at a point?...
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0 votes
3 answers
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Cube root of $-2+i$

Edit: My question comes from finding the solutions of this equation using Cardano's method(because our teacher said :D ): $$x^3-6x+4=0$$ And finally I got: $$x=(\sqrt[3]{2})\sqrt[3]{-2+\sqrt{-1}}⠀+(\...
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6 votes
2 answers
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The Infinitely Nested Radicals Problem and Ramanujan's wondrous formula

In mathematics, a nested-radical is any expression where a radical (or root sign) is nested inside another radical, eg. $\sqrt{2 + \sqrt{3}}$. By extension, an Infinitely nested radical (aka, a ...
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2 votes
0 answers
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Interesting sequences when computing nested square roots

In my previous post, I defined $$f(\color{blue}{m},\color{red}{n})= \sqrt{2\color{blue}{-}\sqrt{2\color{blue}{-}\cdots\color{blue}{-} \sqrt{2\color{red}{+}\sqrt{2\color{red}{+}\cdots\color{red}{+} \...
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6 votes
4 answers
260 views

Simplifying $f(\sqrt{7})$, where $f(x) = \sqrt{x-4\sqrt{x-4}}+\sqrt{x+4\sqrt{x-4}}$

If $f(x) = \sqrt{x-4\sqrt{x-4}}+\sqrt{x+4\sqrt{x-4}}$ ; then $f(\sqrt {7})=\; ?$ I tried solving this equation through many methods, I tried rationalizing, squaring, etc. But after each of them, the ...
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4 votes
1 answer
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How do I simplify $34\csc{\frac{2\pi}{17}}$?

I have $$ 34\csc\left(\dfrac{2\pi}{17}\right)$$ is equal to $$\dfrac{136}{\sqrt{8-\sqrt{15+\sqrt{17} + \sqrt{34 + 6\sqrt{17} - \sqrt{34-2\sqrt{17} } + 2\sqrt{ 578-34\sqrt{17}} - 16\sqrt{34-2\sqrt{17}}...
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1 vote
1 answer
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Finding the value of $\sqrt{z \sqrt{z \sqrt{z}}}...$

I was working on the following nested square root problem: Let $a \in \mathbb R ^+$, what is the value of: $$\sqrt{a \sqrt{a \sqrt{a}}}...$$ I concluded that the answer is $a$ and then I thought ...
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3 votes
2 answers
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$\sqrt{n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}}$

Let $\sqrt{n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}}=x$. Therefore, $$n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}=x^2$$ $$x^2-x-n=0$$ $$x=\sqrt{n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}}=\frac{1 \pm \sqrt{1+4n}}{2}...
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5 votes
0 answers
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How to find the partial derivatives of the following nested expression?

I want to find the partial derivatives of the expression for $v_3(\boldsymbol{u})$ with respect to $u_1$, $u_2$ and $u_3$ from the expressions below. Here $\Phi$ denotes the cumulative distribution ...
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3 votes
0 answers
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$\pi$, from Pentadecagon - infinitely expanding Balloon nested Radical

In this post, I would like to share the findings on derivation of $\pi$ with Pentadecagon inscribed in unit circle. Here the side of each chord is $2\sin12^\circ$ (Bisecting the chord by segment which ...
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3 votes
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First few Prime numbers and nested radicals in association with $\pi$

There is interesting association between $2\cos(\frac{\pi}{60})$ which is $\frac{1}{2}\sqrt{8+\sqrt{15}+\sqrt3+\sqrt{10-2\sqrt5}}$ and first few Prime numbers with infinite expansion of balloon nested ...
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-1 votes
1 answer
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Simplifying $\sqrt{34+15\sqrt2}$ [closed]

$$\sqrt{34+15\sqrt2}$$ If we want $34+15\sqrt2$ to be a nice square $(a+b)^2=a^2+2ab+b^2$, most likely it is the case that $15\sqrt2$ corresponds to $2ab$. I don't know what to do from here. Is there ...
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3 votes
1 answer
122 views

Continued radical of powers of 4 equals 3 [closed]

Can someone explain to me why $$\sqrt{4 + \sqrt{4^2 + \sqrt{4^3 + \sqrt{4^4 + \dots}}}} = 3???$$ I need an answer
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