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, $$x^2-ax+b = 0$$...
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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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### 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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### 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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### 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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### 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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### 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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### $\sqrt{2\sqrt{3\sqrt{4.....\infty}}}$

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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We have $$\frac{1}{4} -\frac{1}{4}{_4F}_3\left({-\frac{1}{5},\frac{1}{5},\frac{2}{5},\frac{3}{5}\atop\frac{1}{4},\frac{1}{2},\frac{3}{4}};1\right) -\frac{1}{\sqrt[4] {5}}{_4F}_3\left({\frac{1}{20},... • 4,081 1 vote 0 answers 111 views ### Show that: A_1A_2=R\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2+...+\sqrt{2}}}}} , the number of square roots being n–1. Let A_1A_2…A_k a polygone inscribed in a circle of radius R, with k=2^n . Show that: A_1A_2=R\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2+...+\sqrt{2}}}}} , the number of square roots being n–1. This ... 2 votes 2 answers 83 views ### Does the 0 solution for \sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x...}}}}} hold any meaning? The function f(x)=\sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x...}}}}}\qquad quickly approaches f(x) =x.$$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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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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### 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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### 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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### 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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### 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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### $\pi$, 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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### 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 ...
### 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 ...
Can someone explain to me why $$\sqrt{4 + \sqrt{4^2 + \sqrt{4^3 + \sqrt{4^4 + \dots}}}} = 3???$$ I need an answer