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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Sum of consecutive square roots inside a square root

$$\large\sqrt{1+\sqrt{1+2+\sqrt{1+2+3+\sqrt{1+2+3+4+\cdots}}}}$$ I saw this somewhere in the internet but, the website didn't provide me any further information. What is the sum of the equation above? What is it called?
Vaishnavi
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Multiple Nested Radicals

$\sqrt{9-2\sqrt{23-6\sqrt{10+4\sqrt{3-2\sqrt{2}}}}}$ I have no idea how to unnest radicals, can anyone help?
suomynonA
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What is this number: $x = \sqrt{1+2^1\sqrt{1+2^2\sqrt{1+2^3\sqrt{1+...}}}}\approx 4.14031...$?

I was watching this Mathologer video about Ramanujan's nested radical identity $$ 3 = \sqrt{1 + 2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}\, , $$ and I decided to look at this for different sets of "radical coefficients". Using powers of $2$ yields the…
John Barber
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Ramanujan's Nested Radical

By noting Ramanujan's Nested Radical, we have $3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$ On the other hand, we can manipulate the number $4$ by applying the similar principle. Here we have $\begin{aligned} 4 & = \sqrt{16} \\ & = \sqrt{1+15}…
Shane Dizzy Sukardy
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Definition of Infinite Nesting?

I have read about nested radicals like $$\sqrt{a+\sqrt{a+\cdots}},$$ and they define the expression as the limit of sequence defined by $a_1=\sqrt a$ and $a_n=\sqrt{a+a_{n-1}}$. Why instead isn't it defined by $f(f(\cdots=x$ iff $f^\infty(d)=x$ for…
Jacob Wakem
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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{align*} E & = \frac{ \sqrt{2} }{ \sqrt{\sqrt[4]{…
QED
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Does a value for $\sqrt{x+\sqrt{x+\sqrt{x+...}}}$ actually exist?

I have seen questions of this type being solved as follows : $\sqrt{x+\sqrt{x+\sqrt{x+...}}}$'s value does not change if we add an $x$ to the expression and square root it. Let the value of this expression be $y$. So $$\sqrt{x+y} = y \implies x+y =…
Rajdeep Sindhu
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Contradicting Nested Radicals

I encountered a question that puzzles me. The task is to find the simplified form of $$x\sqrt[2]{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x\sqrt[7] {x}}}}}}$$ my answer is $$x^\frac{433}{252}$$ but as I graph the unsimplified nested radical to desmos…
rosa
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What is the exact value of $\sqrt{-3+2\sqrt{-5+3\sqrt{-7+4\sqrt{-9+\dots}}}}$

What is the exact value of $$R=\sqrt{-3+2\sqrt{-5+3\sqrt{-7+4\sqrt{-9+\dots}}}}$$ I tried to solve it like $\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$, i.e. I tried to find the sequence function for this expression. I got that for $x=1$ expression…
user164524
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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 $\sqrt{4+\sqrt{18+\sqrt{40+\sqrt{70+\sqrt{108+\sqrt{156+…...}}}}}}$ It is easily converted to the general…
Paul vdVeen
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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 of nested radicals (with a common value of n)…
Will Octagon Gibson
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Ramanujan's Nested Radicals: evaluating $\sqrt{4+\sqrt{16+\sqrt{64+\sqrt{\cdots}}}}$

Find the exact value of $$\sqrt{4+\sqrt{16+\sqrt{64+\sqrt{\cdots}}}}$$ My approach: Suppose $$\sqrt{4+\sqrt{4^2+\sqrt{4^3+\sqrt{\cdots}}}} = p \tag{1}$$ By multiplying each side by $2$, we have $$2\sqrt{4+\sqrt{4^2+\sqrt{4^3+\sqrt{\cdots}}}} =…
Shane Dizzy Sukardy
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evaluating $ \sqrt {2+ \sqrt {3+ \sqrt{4+ \sqrt{5+\cdots}}} }$

How do we evaluate the infinite nested radical $ \sqrt {2+ \sqrt {3+ \sqrt{4+ \sqrt{5+\cdots}}} } $ $\space $? Please help N.B. :- It is not a duplicate
user123733
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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 following result holds true. $$\boxed{x = 3 +…
user983206
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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, $$n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}=x^2$$ $$x^2-x-n=0$$ $$x=\sqrt{n+\sqrt{n+\sqrt{n+...+\sqrt{n+f(n)}}}}=\frac{1 \pm \sqrt{1+4n}}{2}$$ It seems counter-intuitive at first that…
sato
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