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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Equality with nested cubic radicals. Where is my error?
We have
$$\sqrt[3]{ \frac{1 - 3 \sqrt[3]{3}}{11- 3 \sqrt[3]{3}}} = \frac{1 - \sqrt[3]{3}}{2- \sqrt[3]{3}}$$
since
$$\require{cancel}
\sqrt[3]{ \frac{\ 1 - 3^{1/3} 3}{11 - 3^{1/3} 3} }=\frac{(\ 1 - 3^...
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Computationally representing a Fuchsian group
I'd like to learn the math behind this code golf answer, in which the symmetry group of the order-4 pentagonal tiling is represented by integer matrices.
My understanding so far is: In $PSL(2,\Bbb R)$ ...
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Closed form for an alternating values infinite sequence of nested roots
I tried to derive a closed form for an infinite sequence of nested square roots with alternating values and found myself with a 4th degree equation, which I'm fine with, but I was wondering if there's ...
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Ramanujan's nested radical: complex numbers this time
It is well-established that when the square root defined on positive real numbers outputs positive real numbers, the equation below holds:
$$3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$$
for a ...
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Nested Radicals from Brazil
Show that
$$
\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{...
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Sums with nested radicals related to $1/\pi^2$
Context:
I evaluated some sums related to $1/\pi^{2}$ that for a reason I don't know I can't find in the literature.
$$(1)\hspace{.5cm}{1\over 2\sqrt{2}}+{1\over 2^{3}\sqrt{2+\sqrt{2}}}+{1\over 2^{5}\...
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Algebraic methods for evaluating infinite nested square root radicals
Intrigued by an challenge in the Dutch Math Olympiade, I studied a way to evaluate infinite nested radicals.
The question was to evaluate $\sqrt{4+\sqrt{18+\sqrt{40+\sqrt{70+\sqrt{108+\sqrt{156+…...}}}...
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Nested radical question from Harvard MIT maths competition
The question is
$ f(n) = \sqrt{{100}+\sqrt{n}} + \sqrt{{100}-\sqrt{n}} $
What is the minimum value of n for which $f(n)$ is an integer. $n$ is a natural number.
I made an attempt to solve it, and ...
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Geometric series summation with power which is also geometric.
I was just pondering how will this series converge. Assuming $n \ge 0$, find
$$
f(n)=n^{1/2} + n^{1/4} + n^{1/8} + n^{1/16} +\text{...}+ 1
$$
We stop the summation when we hit $1$ i.e. $\lfloor{n^{1/...
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Is this general nested radical for $\pi$ true?
We have,
I. Liu Hui (c. 300 AD)
$$\pi \approx 3\cdot2^{\color{red}8}\times \underbrace{\sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2+\sqrt{\color{blue}1}}}}}}}...
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Effective degree bound for solvability by radicals
Let $P\in{\mathbb Q}[X]$ be an irreducible polynomial of degree $n\geq 3$, and let $\mathbb L$ be the decomposition field of $P$. Denote the Galois group of the extension ${\mathbb L}:{\mathbb Q}$ by $...
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Coefficient cubic for the nested radical equation $\pm \sqrt{n\pm \sqrt{n\pm\sqrt{n\pm \sqrt{n+x}} }}=x$
I read a math article on the net that stated:
For the nested radical equation
$$\pm \sqrt{n\pm \sqrt{n\pm\sqrt{n\pm \sqrt{n+x}} }}=x$$
By repeated squaring we get
$$\left( \left( \left( x^2-n \right)^...
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Proving Rate of Convergence
I am investigating the following coupled sequence:
\begin{align*}
y_0 &= 1\\
x_{n+1} &= \sqrt{1 + \frac{1}{y_n}}\\
y_{n+1} &= \sqrt{1 - \frac{1}{x_{n+1}}}\\
\end{align*}
I am trying to ...
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Is there a name for a radical field extension of the rationals that contains all radical/solvable elements?
I am wondering if there exists a name for the field $F$ such that $\mathbb{Q}\subset F\subset \mathbb{A}$, and $F$ contains all the radical elements such as $\sqrt[7]{2}, \sqrt[3]{3-\sqrt[4]{7}}, \...
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Is there any general way of finding root of a irrational number? (eg, getting $\sqrt2+\sqrt3$ from $\sqrt{5+2\sqrt6}$) [duplicate]
for example:
$(\sqrt2+\sqrt3)^2 = 5+2\sqrt6$ is straightforward.
But how can we get $\sqrt2+\sqrt3$ from $\sqrt{5+2\sqrt6}$?
Is there any general method to do it?
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How does $\sqrt{2+\sqrt{2}}+\sqrt{2-\sqrt{2}}$ become $\sqrt{2(2+\sqrt{2})}$?
I'd like to know how can one simplify the following expression
$$\sqrt{2+\sqrt{2}}+\sqrt{2-\sqrt{2}}$$
into
$$\sqrt{2(2+\sqrt{2})}.$$
Wolfram alpha suggests it as an alternative form, and numerically ...
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Finding the value of $\sqrt[4]{(4+\sqrt7)^{-1}}\sqrt{1+\sqrt7}$ with other approaches
It is a problem from a timed exam,
What is the value of $\sqrt[4]{(4+\sqrt7)^{-1}}\sqrt{1+\sqrt7}$ ?
$1)1\qquad\qquad2)\sqrt[4]2\qquad\qquad3)2\qquad\qquad4)2\sqrt[4]2$
I solved it with two ...
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Ambiguous solution involving radicals
Evaluating the expression: $$\sqrt{5+2\sqrt{6}}+ \sqrt{5-2\sqrt{6}} $$
So my solution involves rewriting the expression inside the bigger radical signs as perfect squares: $$\sqrt{\left(\sqrt{3}+\sqrt{...
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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,
\begin{equation}
x^2-ax+b = 0
\end{equation}...
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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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Simplify $\sqrt[3]{9\sqrt3-11\sqrt2}$
Simplify $$\sqrt[3]{9\sqrt3-11\sqrt2}$$
How can we actually simplify this radical?
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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 ...
user983206
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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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How does $ f(x)=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}. x \ge1 $ produce a piecewise function with an interval that is a constant fucntion?
I found this function in an old math contest.
$$ f(x)=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}},\space \space x\ge 1 $$
This function is identical to the piecewise function,
$$ g(x) =
\begin{cases}
2, ...
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Convergence of Sequence of functions with nested radicals.
I am reading Lascota and Mackey's "Chaos, Fractals, and Noise" (which is great). In an early chapter, they define the following operator (really this is a special case of the Frobenius-...
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Recursive sequence relationship with repeating nested radicals
Here I'd like to introduce:
$6-\sqrt{6-\sqrt{6-\sqrt{6-\sqrt{6}}}}$
If continuing the pattern indefinitely, the limit is $4$. The values are successively higher and lower than $4$.
In a recursive ...
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Is there a general form for infinite nested radicals of degreen n?
Say we have the infinite nested radical
$$\sqrt[n]{a + \sqrt[n]{a + \sqrt[n]{a + \cdots}}}$$
When $n = 2$, this evaluates to $\frac{1 + \sqrt{4a + 1}} {2}$, which is the positive root of the equation $...
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Nested sine and cosine half angle formulas [closed]
I am having trouble with this problem. I know that it is related to the sine and cosine half-angle formulas. I substituted $\frac{\sqrt{3}}{2}$ with $\cos(\frac{\pi}{6})$ and got $\frac{\pi}{24}$ for ...
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Finding a root of a degree 5 polynomial
There are positive integers $m$ and $n$ such that $m^2 −n = 32$ and $$\sqrt[5]{m+\sqrt{n}} + \sqrt[5]{m-\sqrt{n}}$$ is a real root of the
polynomial $$x^5 −10x^3 + 20x −40$$ Find $m+n$.
Okay I write ...
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What is the result of $\sqrt{-\frac{1}{4}+\sqrt{-\frac{1}{4}+\sqrt{-\frac{1}{4}+\sqrt{-\frac{1}{4}+...}}}}$
Given the golden ratio is equal to $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}$ you get $f(x) = \sqrt{1+f(x)}$.
Solving for $f(x)$:
$f(x)^2 = x + f(x)$
$f(x)^2 - f(x) = x$
$f(x)^2 - f(x) + \frac{1}{4} = ...
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A nested radical
Find
$$\large \lim_{n \to
\infty}\sqrt[3]{a_0+\sqrt[3]{a_1+\cdots+\sqrt[3]{a_n}}},$$ where
$\large a_n=6(3^{3^n}+1)^2.$
According to Herschfeld's Convergence Theorem, the limit do exists , since $\...
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Showing that $\sqrt{2+\sqrt3}=\frac{\sqrt2+\sqrt6}{2}$ [duplicate]
So I was playing around on the calculator, and it turns out that $$\sqrt{2+\sqrt3}=\frac{\sqrt2+\sqrt6}{2}$$
Does anyone know a process to derive this?
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Truncation error for $ S = \sum_{m=0}^\infty \frac{\beta^m}{m!}a^{-\frac{m+1}{k}}\Gamma\left(\frac{m+1}{k}\right)$
Define the infinite sums $S_1$ and $S_2$ as
$$ S_1 = \sum_{m=0}^\infty \frac{\beta^m}{m!}a^{-\frac{m+1}{k}}\Gamma\left(\frac{m+1}{k}\right)$$
and
$$S_2 = \sum_{m=0}^\infty (-1)^m \frac{\beta^m}{m!}a^{-...
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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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Solve for $x$ in $\sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x$
Solve for $x$:
$$\sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x$$
I tried to substitute $y=x+2$ and then I try to solve the equation by again and again squaring.
Then I got equation, $$(y-2)(3y^{14}-(y-...
user979855
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nested radical simplification
I'm stuck on this radical simplification $$ (\sqrt{5}+1) \sqrt{10+2\sqrt{5}}$$
In my textbook this product is equal to the $tan(\frac{2\pi}{5})$ and the value is : $$ 4 \sqrt{5+2 \sqrt{5}} $$
I tried ...
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Interrelationship between cyclic infinite nested square roots of 2 and Summation of Geometric progression (partial and infinite)
Disclaimer: - As this is a lengthy post I am going to follow Q & A pattern
Finite and infinite nested square roots of 2 can be solved to some cosine values as discussed earlier here and here
It ...
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Suppose $a_1=1$ and $a_n=\sqrt{n-a_{n-1}}$ $\forall n \in\mathbb{Z_{\ge 2}}$. What is the least integer $n$ for which $a_n>4$? ⌊$a_{100}$⌋?
Suppose $a_1=1$ and $a_n=\sqrt{n-a_{n-1}}$ $\forall n \in\mathbb{Z_{\ge 2}}$. What is the least integer $n$ for which $a_n>4$? What is $\left \lfloor a_{100} \right \rfloor$, where $\left \lfloor u ...
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Infinite nested radicals from Putnam exam 1953
Prove that the following sequence is convergent and find the limit:
$$\sqrt{7}, \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7+\sqrt{7}}}, ... $$
with $x_{n+2}=\sqrt{7-\sqrt{7+x_{n}}}$.
Notice that $$2=\sqrt{7-3}...
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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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simplify radicals
In the book $A=B$ (link) page 11 example 1.5.1 it is written
By setting $\sin a = x$ and $\sin b = y$, we see that the identity $\sin(a + b) = \sin a \cos b + \sin b \cos a$ is equivalent to
$$
\...
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$\sqrt[3]{A+B\sqrt{C}}$ generalized denesting formula
Q: What is the generalized formula for denesting $\sqrt[3]{A+B\sqrt{C}}$?
Recently I've posted a question about nested radicals in solving the cubic equation. I received a comment about an ingenious ...
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does this nested radicle converge? $1+\sqrt{1+\sqrt{\dots}+\sqrt{1+\sqrt{\dots}}}+\sqrt{1+\sqrt{1+\sqrt{\dots}+\sqrt{1+\sqrt{\dots}}}}$
Let $a_0=1$ and $a_n=1+\sqrt{a_{n-1}}+\sqrt{1+\sqrt{a_{n-1}}}$
I want to know if the limit of $a_n$ as n goes to infinity.
$$1+\sqrt{1+\sqrt{1+\sqrt{\dots}+\sqrt{1+\sqrt{\dots}}}+\sqrt{1+\sqrt{1+\sqrt{...
user930105
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Ramanujan's Nested radical Inequality
Recently I was studying functional equation, in which I came across this paragraph about Ramanujan's Nested Radical in Christopher G. Small's book [Functional Equations and How to Solve Them][1].
$$ \...
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When does $r + \sqrt s$ with $r,s \in \mathbb Q$ have a cubic root of the form $u \pm \sqrt v$ with $u,v \in \mathbb Q$?
There are quite a number of related questions in this forum, for example
Denesting Cardano's Formula
Prove that $\sqrt[3]{45+29\sqrt{2}} +\sqrt[3]{45-29\sqrt{2}} $ is rational.
Concering square ...
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Denesting Cardano's Formula
For a depressed cubic equation $x^3 + px + q =0$ having exactly one real root, Cardano's formula gives the real root as $$\sqrt[3]{-\frac{q}{2} +\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} + \sqrt[3]{-\...
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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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