# 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.

691 questions
Filter by
Sorted by
Tagged with
1k views

### Nested radical question (1993 All-Russian Olympiad)

I received this interesting problem in an email. It comes from the $1993$ All-Russian Olympiad grade 10, round 4. Prove that: $$\sqrt{2 + \sqrt[3]{3 + \cdots + \sqrt[1993]{1993}}} < 2$$ I did ...
• 1,809
120 views

• 143
58 views

### Does this ridiculous integral converge?

I was looking through some of my older questions, when I came across this crazy integral I posted. $\int\limits^{2}_{1}\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}dx$ I had approximated it at ...
• 1,597
1 vote
127 views

• 1,673
193 views

• 1,597
89 views

### Nested radicals $\sqrt{C+1+b\sqrt{C+1+b^{2}\sqrt{C+1+b^{3}\sqrt{\cdot\cdot\cdot}}}}=g(x)$ then $\lim_{x\to\infty}(g(x+1)-g(x))=^?1$

Well let the problem first : Conjecture : Let $x>M>0$, $b=\sqrt{x}$,$0<C<1$ then define : $$\sqrt{C+1+b\sqrt{C+1+b^{2}\sqrt{C+1+b^{3}\sqrt{\cdot\cdot\cdot}}}}=g(x)$$ Then it seems we have :...
• 3,630
803 views

• 167
104 views

### Nested Square Roots

Question: Solve for $\sqrt{x\sqrt{x\sqrt{x}}}$ I know this may be a duplicate but I have not found one yet. Additionally I understand the logic for base 8 but why would the answer be $x^\frac 78$ ...
1 vote
205 views

### limit of the sequence $a_n = \sqrt{2+\sqrt{3+\sqrt{2+\sqrt{3+...}}}}$

I have already shown that the sequence is monotone and bounded, so it is convergent. what I need is to calculate the limit when $a_n = \sqrt{2+\sqrt{3+\sqrt{2+\sqrt{3+...}}}}$, $n \rightarrow \infty$, ...
35 views

### What is the rate of convergence of the following sequence (equation with a finite number of nested radicals)?

Let $f(x)=\sqrt{1-x^2}$, $b = 1/\sqrt{2}$. The sequence $(E_n)_{n=1}^{\infty}$ is defined as the solution to the following equation : $$f(E_n - f(E_n -f(E_n - ....-f(E_n - b)))) = E_n -1,$$ where the ...
• 417
1 vote
101 views

### Can this set of equations be extrapolated to a complete pattern?

Quick Background We have five independent variables that can each be any real number greater than zero: $d_{max}$ $v_{max}$ $a_{max}$ $j_{max}$ $s_{max}$ These variables are linked to a chain of ...
• 1,673
1 vote
121 views

### How to simplify the cubic radical $\sqrt[3]{a\sqrt{b}-c}$?

How to simplify the cubic radical $\sqrt[3]{a\sqrt{b}-c}$ ? So I encountered a particular problem in chapter of surds and radicals to find the cube root of $38\sqrt{14}-100\sqrt{2}$ . So I took out 2√...
• 642
132 views

### An Olympiad Question on cube roots. [closed]

If $$\sqrt[3]{\sqrt[3]{2}-1} = \sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{\frac{1}{9}}$$where $a$ is greater than $b$, then find the value of $a+2b+3$. I got this question on an olympiad handout and couldn't ...
52 views

### An Infinite Sum of Nested Radicals (With A Partial Solution)

Define $f(y) = \sqrt{y^{2}+\frac{2x}{N}+-\frac{2yx}{N}}$, where $x$ is any positive real number and $N$ is the number of times we wish to compose $f$ with itself. I wish to find the limit as the ...
63 views

• 61
207 views

• 54.5k
53 views

1 vote
77 views

### 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?
• 31
245 views

### 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 ...
• 51
### 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 ...