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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Infinite Nested Radical

Ramanujan discovered that $$x+n+a=\sqrt{ax+(n+a)^2+x\sqrt{a(x+n)+(n+a)^2+...}}$$ (see equation (27) here). I didn't understand how we can use this (basically what to put in place of $x, n, a$) to calculate the value of…
sato
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Find the value of $\sqrt{1+\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+\cdots}}}}}$ .

We have: $\sqrt{(1+0)+\sqrt{(4+1)+\sqrt{(9+2)+\sqrt{(16+3)+\sqrt{(25+4)+\cdots}}}}}.$ Basically I'm not getting any clue at the moment for reducing the infinite nested radicals. Any hint would be helpful. Thanks in advance.
Dhrubajyoti Bhattacharjee
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How to simplify this:$\sqrt{5\sqrt{3}+6\sqrt{2}}$?

How to simplyfy this:$\sqrt{5\sqrt{3}+6\sqrt{2}}$. I know I should use nested radicals formula but which one is $A$ and $B$.Using the fact $A>B^2$ you can find $A$ and $B$. But $C^2=A-B^2$ isn't a rational number then we have again a nested…
Taha Akbari
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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)}\\…
MathAdam
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Proving The Nested Radical

For $a>b^2$, prove that $\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}} = \sqrt{a-\dfrac34b^2}-\dfrac12b$. Attempt: After assuming the value of the nested radicals to be $S$, I got $$S = \dfrac{\left(\dfrac{a-S^2}{b}\right)^2-a}{b},$$ but now I…
Shane Dizzy Sukardy
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Why $\sqrt{23-\sqrt{17}}-2\sqrt{7-\sqrt{17}}=\sqrt{71-17\sqrt{17}}$ is true?

Easy to show this identity after squaring twice of the both sides. But why it turned out true? For example, if we want to prove that $$\sqrt{23-3\sqrt{5}}-2\sqrt{3-\sqrt{5}}=\sqrt{3+\sqrt{5}},$$ we can do it without…
Michael Rozenberg
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Denest $\sqrt{5^{\frac{1}{3}}-4^{\frac{1}{3}}}$

I'd like to ask how to denest the nested radical $\sqrt{5^{\frac{1}{3}}-4^{\frac{1}{3}}}$?
Hang Wu
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Ramanujan's Infinite Root

While I was surfing on the Internet yesterday, I watched a video about Ramajuan's infinite root. After that I had tried on my own and I got the point. $ 3=\sqrt9$ $3=\sqrt{1+8}$ $3=\sqrt{1+2 \cdot 4}$ $3=\sqrt{1+2\cdot \sqrt{16}}$ …
Alper
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Does this radical ${a-1\over 2}\left({\sqrt[3]a+1\over \sqrt[3]a-1}-1\right)=...$ hold?

Consider the nested radical $(1)$ $${a-1\over 2}\left({\sqrt[3]a+1\over \sqrt[3]a-1}-1\right)=2\sqrt[3]a+\left[(a^2-7a+1)+(6a-3)\sqrt[3]a+(6-3a)\sqrt[3]{a^2}\right]^{1/3}\tag1$$ An attempt: $x=\sqrt[3]a$ $$\left({a-1\over 2}\left({x+1\over…
gymbvghjkgkjkhgfkl
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Variation to Ramanujan's infinite nested radical with primes

Ramanujan's infinite nested radical states that $$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}=3.$$ Now instead, consider the following infinite nested radical $$\sqrt{1+p_1\sqrt{1+p_2\sqrt{1+p_3\sqrt{1+\cdots}}}}$$ where $p_n$ represents the…
user51819
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Find the value of $\sqrt{6+\sqrt{6+\sqrt{6+\dots}}}$

Evaluate $$\sqrt{6+\sqrt{6+\sqrt{6+\dots}}}$$ I need some help with this question because I have no idea what is going on and help would be greatly appreciate :)
Help
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What is value of $\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\ldots}}}}=$?

My question: how to find nested radical having $n$th roots $$\large\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\ldots}}}}=?$$ My try: $$\large\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\ldots}}}}=y$$ $$\large\sqrt[n]{xy}=y$$ $$\large xy=y^n$$$$…
user766881
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Another way to evaluate the nested radical $x=\sqrt{2+{\sqrt{2+\sqrt{2+\ldots}}}}$

Consider the following expression: $$x=\sqrt{2+{\sqrt{2+\sqrt{2+\ldots}}}}$$ It can easily be evaluated by realizing that: $$x=\sqrt{2+x}$$ After squaring both sides and solving the quadratic equation, we find that $x=2$. But my question is: out of…
user263286
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How to show that some nested radicals simplify to other

Show that $\sqrt{8}\cdot\sqrt{9-\sqrt{77}}=2\cdot\sqrt{11}-2\cdot\sqrt{7}$ I have tried multiplying the radicals but that didn't work. The resulting radicals do not add up or get subtracted. I have tried taking commons also but that also doesn't…
Hamza
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Find all x where that limit is true

Let the following expression, with $n \in \mathbb{N}$ $$ T_n = \underbrace{\sqrt{x \sqrt{x \sqrt{x \dots \sqrt{x}}}}}_{\text{n times}} $$ It's easy to see that $$\lim_{n \to \infty} T_n = x$$ Find all x where that limit is true
Gustavo Viana
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