Mathematics Stack Exchange
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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.

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. Reference: Wikipedia

Some nested radicals can be rewritten in a form that is not nested. Rewriting a nested radical in this way is called denesting.

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Square root till infinity

What is the value of $\sqrt{x + \sqrt{ x + \sqrt{ x + \cdots } } }\,$? I know the basic trick to calculate this using $f = \sqrt{ x + f }$. But, I want more accurate answer which is I am not getting with this formula.
Vaishal
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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 this topics so how can I use polynomial nesting…
tent123
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Alternate method to solve $\sqrt{11\sqrt{11\sqrt{11...4\, \text{times}}}}$

Question : What is the value of $$\sqrt{11\sqrt{11\sqrt{11...4\,\text{times}}}}$$ I did it by solving square root one by…
Wolgwang
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Existence of nested square roots start and end and in between radicals inf

Do we have nested square roots with initial and final term and infinite terms in between? For example $$\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{3}}}}}$$ which happens in modified Viète nested radical Or nested radicals like general form as…
Sivakumar Krishnamoorthi
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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 solutions when solving for $x$? For example, you could…
S.C.
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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 $(x-1)^2 = 2$?
yena shah
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calculate the limit for $\sqrt{1 + \sqrt{2 + \sqrt{1 + \sqrt{2 + ......}}}}$

What I got right now is: $a_0 = \sqrt{1 + \sqrt{2}}$, $a_1 = \sqrt{1 + \sqrt{2 + a_0}}$, and $a_n = \sqrt{1 + \sqrt{2 + a_{n-1}}}$. I can say $a = \sqrt{1+\sqrt{2 + a}}$, right? If yes, how to show that $a = \sqrt{1+\sqrt{2 + a}}$?
Vincent Zheng
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Denesting a nested radical where $x,y\in\mathbb{Z}$

Let $x$ be an integer. If $$\sqrt{x+\frac 12\sqrt{2011}}-\sqrt{x-\frac 12\sqrt{2011}}=y\tag{1}$$ Where $x,y\in\mathbb{Z}$, then find the value of $x$ The way I solved it was simply moving one radical to the right hand side and repeatedly squaring…
Frank
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Prove $\frac{\sqrt[n]k+1}{\sqrt[n]k-1}=\frac{2}{k-1}\left(\frac{k+1}{2}+\sum_{j=1}^{n-1}\sqrt[n]{k^j}\right)$

Ramanujan's nested radical (1) $$\frac{\sqrt[4]5+1}{\sqrt[4]5-1}=\frac{1}{2}\left(3+\sqrt[4]5+\sqrt[4]{5^2}+\sqrt[4]{5^3}\right)$$ We proposed the general formula of Ramanujan's equation…
user339807
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A sequence of nested radicals and its limit

Please Help me in the following Problem What is the Number Of Natural Numbers ,$n\le30$ for which $\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}$ is also a prime number. The only way I am able to find to solve this is calculate each and every term once but it…
S.Bansal
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What is the value of $\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{8}+\sqrt{\frac{1}{128}+...}}}}$?

An interesting radical I came up with and I'd like to know your approach. Each term is of the form: $\frac{2^{2^r}}{2^{2^{r}+2^{r-1}+...+2+1}} = \frac{1}{2^{2^{r-1}+...+2+1}}$
Joshua Farrell
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Solving a radical to find the value

How to find the value of this nested radical expression ? $$\sqrt{7+\sqrt{7-\sqrt{7+\sqrt{7-\cdots\infty}}}}$$
atulvinod1911
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