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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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Confusing regarding a nested radical equation

For all $a\in\Bbb R$ solve the equation $$\sqrt{x^2+4a^2\sqrt{x+a}}=x+2a$$ It is immediate to see that we got the restriction $x\geqslant-a$ (even though not given I assume that this equation is ...
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1answer
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Simplify $\sqrt {9 + 2(1 + \sqrt {3})(1 + \sqrt {7})}$

Simplify $\sqrt {9 + 2(1 + \sqrt {3})(1 + \sqrt {7})}$ I know this requires denesting but I don't know how to begin here.
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Problem with $\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}$

How to simplify $$\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}?$$ Rationalise the denominator $$\frac{\sqrt{6+4\sqrt{2}}}{4}(2-\sqrt{2})$$ This is still not simplify.
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Four Nested Radicals

For how many positive integers $n$ is $$\sqrt {n+\sqrt {n+\sqrt {n+\sqrt {n}}}}$$ an integer? I still have yet to find a single integer that satisfies this (there's some that get extremely close, ...
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$\lim\limits_{n \to \infty}\sqrt{1+\sqrt{\frac{1}{2^2}+\sqrt{\frac{1}{3^2}+\cdots+\sqrt{\frac{1}{n^2}}}}}.$

Problem Evaluate $\lim\limits_{n \to \infty}T_n$ where $$T_n=\sqrt{1+\sqrt{\frac{1}{2^2}+\sqrt{\frac{1}{3^2}+\cdots+\sqrt{\frac{1}{n^2}}}}}.$$ Analysis It's obvious that $T_n$ is increasing with a ...
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How to use the formula in negative roots?

This formula for nested-radicals: $\sqrt{A+\sqrt{B}}=\sqrt{\frac{A+C}{2}}+\sqrt{\frac{A-C}{2}}$, works fine with $B > 0$, and with $A + \sqrt{B} > 0$, but doesn't work with complex numbers. I ...
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Find $\sqrt{a} + \sqrt{b}\space$ as an exact answer for $\{a,b\} \in \mathbb R^+$

(Sorry if my MathJax is strange, I just skimmed through the tutorial and tried to make it work) I want to find what is basically a sum formula for square roots, similar how it exists for $\log(a) + \...
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On two nested radicals and divisibility

The last days I was playing around with two nested radicals which, as I learned here, can be simplified: $$u(x) =\sqrt{x + \sqrt{x +\sqrt{x +\sqrt{x +...}}}} = \frac{1}{2}(1+\sqrt{1+4x})$$ $$l(x) = \...
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1answer
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Convergence of $\sqrt[n]{(1+x)^n-(1+x)+\sqrt[n]{(1+x)^n-(1+x)+\sqrt[n]{\cdot\cdot\cdot}}}$

If one writes $$1+x=\sqrt{(1+x)^2}=\sqrt{1+2x+x^2}=\sqrt{x+x^2+(1+x)}$$ then one has a recursive definition of the function $1+x$ which can be used to write $1+x$ as the infinite nested radical: $$1+x=...
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How to simplify $\sqrt{2+\sqrt{3}}$ $?$

Simplify $\left(\frac{2(\sqrt2 + \sqrt6)}{3(\sqrt{2+\sqrt3}}\right)$ The answer to this question is $\frac{4}{3}$ in a workbook. How would I simplify $\sqrt{2+\sqrt3}$ $?$ If it was something like $...
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What is $\lim\limits_{n\to \infty} \sqrt{\frac1{1^2}+\sqrt{\frac1{2^2}+\sqrt{\frac1{3^2}+\cdots+\sqrt{\frac1{n^2}}}}}$? [closed]

By numerical calculations we yields 1.466723564..., from which I can't see any useful properties of that value. The same problem can be found at https://www.quora.com/What-is-the-value-of-sqrt-1-sqrt-...
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Prove value of $\sin(15°)$

I'm doing the following exercise: prove that $$ \sin(15°)=\frac{1}{2\sqrt{2+\sqrt{3}}} $$ I'm using this formula: $$ \sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b) $$ to get: $$ \sin(45-30)=\sin(45)\cos(...
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Infinitely Nested Radical with Fibonacci Coefficients

I was wondering if the following infinitely nested radical can be evaluated. $x= \sqrt{1+ \textbf{1}\sqrt{1+ \textbf{1}\sqrt{1+ \textbf{2}\sqrt{1+ \textbf{3}\sqrt{1+ \textbf{5}\sqrt{1+ \dots }}}}}} $ ...
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Ramanujan's radical and how we define an infinite nested radical [duplicate]

I know it is true that we have $$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}=3$$ The argument is to break the nested radical into something like $$3 = \sqrt{9}=\sqrt{1+2\sqrt{16}}=\sqrt{1+2\sqrt{...
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1answer
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Prove that inequality $\sqrt{2\sqrt{4\sqrt{8…\sqrt{2^n}}}} \leqslant n+1$

Let $n$ be the integer. Prove that $$\sqrt{2\sqrt{4\sqrt{8....\sqrt{2^n}}}} \leqslant n+1$$ SOURCE: BANGLADESH MATH OLYMPIAD I am a new beginner at the infinite radical and sequence. I don't know ...
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Proof that 2*2/sqrt(2)*2/sqrt(2+sqrt(2))*2/sqrt(2+sqrt(2+sqrt(2)))*… equals PI?

I found this formula that $\pi=2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}\cdot\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdot...$ I tested it out and it seems to be true, but I don't get why ...
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2answers
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How can I simplify $\sqrt{\frac{5+\sqrt{5}}{2}}$?

I've tried to see the root as $\sqrt{\frac{5+\sqrt{5}}{2}} = \sqrt{a}+\sqrt{b},$ but this method doesn't give me something good.
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3answers
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How do I simplify $\sqrt {4(2- \sqrt{3})}$ into $\sqrt{6} - \sqrt{2}$

This might be a stupid question, but how do I get from $$\sqrt {4(2- \sqrt{3})}$$ to $$\sqrt{6} - \sqrt{2}$$ It is obvious if you squared both, they both equal $8 - 4 \sqrt{3}$, but I'm wondering how ...
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Finding rational numbers in an equation with two variables

How should we find two rational numbers $\alpha$, $\beta$ such that $\sqrt[3]{7+5\sqrt{2}}=\alpha+\beta\sqrt{2}$? The answer I got alpha = 1 and betta = 1. If I'm wrong, please correct me. Thank you
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Tough Irrational Equation highschool

Have been trying to solve this irrational equation for a day but as it seems, i'm not going anywere with it. Can somebody offer me a tip ? Thanks! *Tried a "t" substitution for x squared but it still ...
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1answer
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Is it guaranteed that $\sqrt[3]{a+\sqrt{b}}$ can be denested with or without complex numbers?

I tried to use the cubic formula before, but was always stuck at simplifying the cube root. I had learned that you can always simplify $\sqrt{a+\sqrt{b}}$ by solving, but can I always simplify $\sqrt[...
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3answers
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Why does infinitely repeating $\sqrt[n]{x}$ converge to 1? [closed]

A coworker showed me an interesting fact of math this morning. He stated that if you get the square root of $x$ and then get the square root of the result, and so on, and so forth, to infinity, then ...
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0answers
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What is the minimum value of $f_\infty=\frac{x}{\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\cdots}}}}$?

In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers. What is the minimum value of $$f_\...
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0answers
66 views

Solving $\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}$ [duplicate]

$$\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}$$ My approach 1st approach: Let $\sqrt{1+\sqrt{2+a}} = x$, where $a$ is a result of the lower portion of the continued nested radical and $1+\sqrt{2+a}$ is a ...
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Is there a nice algorithm for testing the equality of two nested radicals?

I'm talking about a computer algorithm in the realm of symbolic computation. I also need to be able to write the code myself as opposed to using a third-party tool. One way I thought of was to find ...
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885 views

What is the maximum value of this nested radical?

I was experimenting on Desmos (as usual), in particular infinite recursions and series. Here is one that was of interest: What is the maximum value of $$F_\infty=\sqrt{\frac{x}{x+\sqrt{\frac{x^2}{x-...
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3answers
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Does a given infinite nested radical have infinitely many solutions?

Given a standard infinitely nested radical such as: $$x = \sqrt{1 + \sqrt{1 + \sqrt{ 1 + ...}}}$$ depending on where you choose to first substitute $x$ in the nest, aren't there infinitely many ...
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1answer
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How would you prove $\int^8_0\frac1{\sqrt{x+\frac1{\sqrt{x}}}}dx<4-\frac1{2019}$?

We would like to prove the following inequality. $$\int^{8}_{0}\frac{1}{\sqrt{x+\frac{1}{\sqrt{x}}}}\,dx<4-\frac{1}{2019}\tag{1}$$ What I've tried is using the AM-GM inequality, $$x+\frac{1}{\...
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How to prove the nested sets be the definition of the indexed collection of sets

enter image description here enter image description here I only know the definition of the indexed collection sets, but I dont know how to prove the nested sets by this definition. Can anyone help me?...
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1answer
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What's equal the below power nested radical?

it is well known that $$\frac{2}{\pi}=\sqrt{\frac12}{\sqrt{\frac12+\frac12\sqrt{\frac12}}{\sqrt{\frac12+\frac12\sqrt{\frac12+\frac12\sqrt{\frac12}}}{\sqrt{\frac12+\frac12\sqrt{\frac12\cdots}}}}}$$ ...
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Can we make it equal to x?

$\sqrt{6 +\sqrt{6 +\sqrt{6 + \ldots}}}$. This is the famous question. I have to calculate it's value. I found somewhere to the solution to be putting this number equal to a variable $x$. That is, $\...
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2answers
47 views

How to solve this radical expression

I've been trying to solve this expression for at least two hours now... And I always get stuck towards the end, I don't know what I'm missing. $\frac 1{xy} \times (\sqrt{xy} - \frac{xy}{x-\sqrt{xy}})\...
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4answers
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How to prove $\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\ldots}}}}}}=\frac {\sqrt {4a-3}-1}2$

So, I was watching this video by blackpenredpen where he mentions that $$\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\ldots}}}}}}=\frac {\sqrt {4a-3}-1}2$$ so I wanted to try and prove it ...
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0answers
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How to simplify nested radicals?

How to simplify this expression?$$\sqrt{\smash[b]{18+\sqrt{260}}}-\sqrt{\smash[b]{12+\sqrt{140}}}-\sqrt{\smash[b]{20-2\sqrt{91}}},$$ which equals $0$. But how do I prove it? My attempt \begin{align}\...
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1answer
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How can I figure out a nested radical?

I have to find the value of: ( The picture ) And I have this solution: Now, I understood how they took $x = \sqrt{1+2x}$ $\implies x^2-2x-1= 0$ But how did they take $(x-1)^2 = 2$?
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Simplify $\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$

Simplify $$\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$$ Found in a book with tag "Moscow 1982", the stated answer is $1+\sqrt[4]{5}$. Used all tricks that I know but without success. ...
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2answers
215 views

Find $\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{a}}{a}}}{a}}}{\begin{array}{c} a\\\vdots \end{array}}}$

Assuming $m\in \Bbb N\setminus\{0,1\}$ and $a\in \Bbb R_+\setminus\{0\}$, find $$\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{a}}{a}}}{a}}}{\begin{array}{c} a\\\vdots \end{array}}}$$ In a ...
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1answer
326 views

Show that $\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+…}}}}=\sqrt{a-\frac{3b^2}{4}}-\frac{b}{2}$

Assuming that $a>b^2$ show that $$\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+...}}}}=\sqrt{a-\frac{3b^2}{4}}-\frac{b}{2}$$ (corrected) This problem listed in a contest-math preparation book with the ...
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3answers
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If $x=-1/4$ does $S=\sqrt{x+\sqrt{x+\sqrt{x+…}}}=1/2$?

If $x=-1/4$ does $S=\sqrt{x+\sqrt{x+\sqrt{x+...}}}=1/2$? My attempt: As $S^2=x+S$, it holds that $S=\frac{1\pm \sqrt{(-1)^2+4x}}{2}$, which has a real solution for $x\ge -1/4$. Therefore, if $x=-1/...
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Showing that $\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$?

$$\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$$ In the second nested radical, the repeating pattern is $(-,-,+)$. I approached this ...
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1answer
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How we can show this: $\sqrt{3+\sqrt{3}}-\sqrt{3-\sqrt{3}}=\sqrt{6-2\sqrt{6}}$

Only way in my mind to show that : $$\sqrt{3+\sqrt{3}}-\sqrt{3-\sqrt{3}}=\sqrt{6-2\sqrt{6}}$$ is to multiply $\sqrt{3+\sqrt{3}}-\sqrt{3-\sqrt{3}}$ by the conjugate factor which is $\sqrt{3+\sqrt{3}}+...
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3answers
91 views

Solve this inequality with nested radicals (possibly by induction)

I tried to solve this problem by induction but didn't succeed. Given the series $$ a_n = \sqrt{1+\sqrt{2+\sqrt{3+\sqrt{... + \sqrt{n}}}}}$$ Prove that $a_n < 2 (\forall n \in \mathbb{N^*}) $ ...
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2answers
68 views

How to find integers $p$ and $q$ such that $(p\sqrt{2}+q)^2=34-24\sqrt{2}$

Find integers $p$ and $q$ such that $(p\sqrt{2}+q)^2=34-24\sqrt{2}$. I approached this question first by expanding the the left-hand side to get: $$2p^2 +2\sqrt{2}pq+q^2 = 34-24\sqrt{2}$$ The ...
3
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1answer
200 views

Recursive square root inside square root problem

I have been debating this issue for days: I can't find a recursive function of this equation: $\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}$ has been trying to find a solution this ...
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0answers
20 views

Is there a systematic method to simplify complicated multi-nested radicals?

I am currently working on a project in which I use symbolic computer algebra to demonstrate the Schläfli symbol of regular polyhedra. The program I wrote is correct when computing the vertex ...
6
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0answers
128 views

Show that $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4…}}}$ [duplicate]

I was solving a problem, asking me to prove the identity $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4...}}}$ which was first posed by Ramanujan. The standard answer goes as $$3=\sqrt{1+2\cdot4}=\sqrt{1+2\sqrt{1+3\...
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3answers
351 views

How to prove that $\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}=4$?

Using the Cardano formula, one can show that $\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}$ is a real root of the depressed cubic $f(x)=x^3-6x-40$. Actually, one can show by the calculating the ...
17
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4answers
1k views

How to simplify $\int{\sqrt[4]{1-8{{x}^{2}}+8{{x}^{4}}-4x\sqrt{{{x}^{2}}-1}+8{{x}^{3}}\sqrt{{{x}^{2}}-1}}dx}$?

I have been asked about the following integral: $$\int{\sqrt[4]{1-8{{x}^{2}}+8{{x}^{4}}-4x\sqrt{{{x}^{2}}-1}+8{{x}^{3}}\sqrt{{{x}^{2}}-1}}dx}$$ I think this is a joke of bad taste. I have tried every ...
4
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6answers
777 views

Having trouble simplyfing radicals of this sort

I'm studying radicals and rational exponents. I'm having lots of hardships with problems of this sort: prove $$\sqrt{43+24\sqrt{3}}=4+3\sqrt{3}$$ I keep going around and around experimenting with ...
20
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2answers
475 views

Solving the infinite radical $\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+…}}}}$

$$\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+\cdots}}}}$$ This is a modification on the well-known Ramanujan infinite radical, $\sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}}$, except it cannot be solved by the ...