For questions involving radical numbers or expressions (i.e. expressions which involve $\sqrt[n]{\text{something}}$).

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-1
votes
1answer
41 views

Find the limits : $\lim_{x\to\infty} x\left(\sqrt{x^2+2x}-2\sqrt{x^2+x}+x\right) $ [on hold]

$$\lim_{x\to\infty} x\left(\sqrt{x^2+2x}-2\sqrt{x^2+x}+x\right) =\ ? $$
0
votes
1answer
29 views

Finding a rule for the outcome of $\sqrt[4]{r}(a+\sqrt{r})$

It is well known that $$\sqrt{4+3\sqrt{2}}=\sqrt[4]{2}(1+\sqrt{2})\tag{1}$$, and similarly, $$\sqrt{10+6\sqrt{5}}=\sqrt[4]{5}(1+\sqrt{5})\tag{2}$$$$\sqrt{6+4\sqrt{3}}=\sqrt[4]{3}(1+\sqrt{3})\tag{3}$$$$...
-2
votes
1answer
119 views

$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$ change a sign to be rational [on hold]

I have this problem: $$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$$ Prove that you need to change ONLY a sign (to convert a $+$ to $-$) of a single square root, for the sum to be rational. EDIT:...
2
votes
3answers
45 views

if $x \neq 0$ and $x = \sqrt{4xy - 4y^2}$, then how does expressing $x$ in terms of $y$ mean $x = 2y$?

I have the equation $x=\sqrt{4xy - 4y^2}$, and I know that $x=2y$ when expressed in terms of $y$, but I'm not sure of the process to get there. I know that \begin{align}\sqrt{4xy - 4y^2} &= \...
0
votes
1answer
11 views

Unchanged Conjugate Radical? (Rationalizing Demoninators)

I'm on one of the more difficult practice problems on Excercise 1-6 in "AoPS:Vol. 1". Problem £4 : Ex. 1-6. The hint details that we should multiply $$\left(\frac{1}{\sqrt 1 \sqrt 2}\right) $$ by it'...
0
votes
4answers
53 views

If $x$ is positive, then why does $\frac{1}{\sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} - \sqrt{x}$?

Given that $x$ is positive, $\frac{1}{\sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} - \sqrt{x}$ I've been trying to convert the left side of the equation to the right side: $$ \frac{1}{\sqrt{x+1} + \sqrt{x}}$...
0
votes
2answers
37 views

Rationalizing Denominator w/ a radical

Reading "Art of Problem Solving : Vol. 1". Stuck on Excercise 1-6 : $$ {\sqrt2\over \sqrt6-2} $$ I know, we must rationalize, multiplying by $$ { \sqrt6+2\over \sqrt6+2} $$ However, what would ...
2
votes
1answer
74 views

A field of Radical Sums

I am dealing with a computation that yields numbers that are sums of radicals of the following form: $N=\sum_{i=0}^{m}{a_i\sqrt{b_i}}$ Where $a_i,b_i \in \mathbb{Q}$ (rationals). The context is ...
4
votes
1answer
75 views

Is there an 'interesting' way to derive this expression?

So I was asked to prove the following term is equal to $2016$: $$ \left( \frac{251}{ \frac{1}{ \sqrt [3] {252} - 5 \sqrt [3] {2} } -10 \sqrt [3] {63} } + \frac {1} { \frac {251} { \sqrt [3] {252} +5 ...
1
vote
1answer
29 views

complex modulus and square root

I am failing to understand something about complex square roots: If we fix the argument $\theta\in(0,2\pi],$ that is we take the positive real line as branch cut, than for $z=r\mathrm{e}^{i\theta}$, $...
2
votes
0answers
36 views

Minimize a huge two-variable logarithmic-trigonometric-radical expression (MSU entrance early July 2016)

Minimize \begin{align}R(a,x)&=\sqrt{13+\log_a\left(\cos\left(\frac xa\right)\right)^2+\log_a\left(\cos\left(\frac xa\right)^4\right)}+\sqrt{97+\log_a\left(\sin\left(\frac xa\right)\right)^2-\...
1
vote
1answer
75 views

Square root inequality revisited

This is a follow-up question of this one: Proof of the square root inequality I am interested in the following generalizations of the square root inequality. Let $\varepsilon,\delta>0.$ Then $$\...
7
votes
6answers
1k views

Proof of the square root inequality [duplicate]

I stumbled on the following inequality: For all $n\geq 1,$ $$2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}.$$ However I cannot find the proof of this anywhere. Any ideas how to ...
3
votes
1answer
55 views

Simplifying $\sqrt[5]{1+g+g^3}=\frac {\sqrt{1+g^2}}{\sqrt[10]{5}}$ and similar ones

I saw that Ramanujan simplified many radicals such as: For $g^5=2$ $$\sqrt[5]{1+g+g^3}=\frac {\sqrt{1+g^2}}{\sqrt[10]{5}}\tag{1}$$ For $g^4=5$ $$\frac {\sqrt[5]{3+2g}-\sqrt[5]{4-4g}}{\sqrt[5]{3+2g}+\...
7
votes
9answers
263 views

How do I prove that $\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$

How do I prove that $$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$ without using the calculator?
56
votes
3answers
4k views

Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots ...
2
votes
2answers
105 views

Find an irrational number $n$ such that $n^n$ is a rational number.

Find an irrational $n$ such that $n^n$ is a rational number. I have some tries to find this... I have tried so much numbers but no success. How can I find them.
4
votes
1answer
89 views

Necessary and sufficient condition that $ \lceil \sqrt { \lfloor x \rfloor } \rceil = \lceil \sqrt { x } \rceil $

I am stumped at the following paragraph, which comes from Concrete Mathematics, Chapter 3, Section 2, Page 73: What is a necessary and sufficient condition that $ \lceil \sqrt { \lfloor x \rfloor }...
2
votes
2answers
60 views

How do you say: $\sqrt[z] x$ where $z > 3$?

My whole mathematics is in chaos right now.... I forgot how to say: $\sqrt[z] x$ and I don't know where else to ask - I know how to say ${d}\sqrt x$ - this is just: $d$ times the square root of $x$; ...
0
votes
0answers
28 views

Plus or Minus (Square Root)

I was evaluating $$\lim _{(x,y,z) \to (2,8,1)} \sqrt{xy} \tan \frac {3 \pi z} 4$$ and got as far as $\sqrt 4 \cdot (-1)$. Now, how do I know if it is a positive or negative root besides graphing it ...
2
votes
1answer
37 views

A square-root approximation method that would halt on $\sqrt{378}$

Back in the early $'90$s, I used to program in a (now obsolete) scripting language called LOGO. Now, one peculiar glitch that I encountered at the time, was the interpreter halting on $\sqrt{378}$. ...
-1
votes
0answers
16 views

Potential costing [on hold]

I would like to know what does it mean by potential price and how to get a potential price. Here is a sample problem. A man specializes in the sale of vintage cars. For cars more than ten years old, ...
2
votes
1answer
116 views

Points on the elliptic curve for Ramanujan-type cubic identities

Given the rational Diophantine equation, $$t^3 - t^2 - \tfrac{1}{3}(n^2 + n)t - \tfrac{1}{27}n^3=w^3\tag1$$ Two points are, $$t_0 = 0\tag2$$ $$t_2 = \frac{-(1 + 2 n) (1 + 11 n + 42 n^2 + 14 n^3 + 13 ...
0
votes
3answers
63 views

How to find an exact limit of a function which is subtracting square roots of quadratic equations?

Find exact value of limit: $$ \lim \limits_{x \to \infty} \sqrt{( 3x^2 + 8x + 6)}-\sqrt{( 3x^2+3x+4)} $$ Here is what I've got so far: $$\lim \limits_{x \to \infty} \frac{(\sqrt{( 3x^2 + 8x + 6)}-\...
2
votes
3answers
127 views

What are the differences between: $\sqrt{(-3)^2}$, $\sqrt{-3^2}$ and $(\sqrt{-3})^2$. [closed]

First, is $\sqrt{-3}$ is equal to $-3$ or is it imaginary? What is the difference between: $\sqrt{(-3)^2}$ $\sqrt{-3^2}$ $(\sqrt{-3})^2$ Can I write $(\sqrt{-3})^2 = -3$? And, given the rule ...
0
votes
1answer
40 views

square root of sum vs. sum of quare roots for a certain form

Given my original formula $\sqrt { \left( 1-a-b \right) \left( 1+c+d \right) } $, i notice that it is approximately equal to $\sqrt { \left( 1-a \right) \left( 1+c \right) } +\sqrt { \left( 1-b \...
7
votes
0answers
119 views

Generalizing Ramanujan's cube roots of cubic roots identities

(This extends this post.) Define the function, $$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$ where the $x_i$ are roots of the cubic, $$x^3+ax^2+bx+c=0\tag2$$ While $G(t)...
2
votes
1answer
48 views

Does there exist an algebraic solvability algorithm?

I was ruminating over quintics and got curious about the following idea. Consider a quintic equation: $$ Q(x) a_0 + a_1 x + a_2 x^2 + ... a_5 x^5$$ Such that the solutions to $$ Q(x) = 0 $$ Are ...
10
votes
2answers
264 views

What's the explanation for these (infinitely many?) Ramanujan-type identities?

Define the function, $$F(\beta) := \sqrt[3]{\beta+x_1}+\sqrt[3]{\beta+x_2}+\sqrt[3]{\beta+x_3}\tag1$$ where, $$x_1 =2\cos\big(\tfrac{2\pi }{7}\big),\;x_2 =2\cos\big(\tfrac{4\pi }{7}\big),\; x_3 = ...
-1
votes
1answer
38 views

Solution of the inequality $2ec{\sqrt{ad}}\lt dc^2+ae^2$

I just want to find a way to prove that the inequality $2ec{\sqrt{ad}}\lt dc^2+ae^2$ is true because I need it for a prove. Thanks for your help!
2
votes
1answer
114 views

The $n$-th root of $(1+q^n)^2$ is irrational

Let $0<q<1$ be rational. I am suspecting that $\sqrt[n]{(1+q^n)^2}$ is irrational. Can someone please help me to prove or to disprove this? $n=1$ and $n=2$ are simple cases. I am interested ...
11
votes
5answers
360 views

Subtracting expressions with radicals

I want to subtract the expressions $20\sqrt{72a^3b^4c} - 14\sqrt{8a^3b^4c}$. I simplified this to $120ab^2\sqrt{2ac}-28ab^2\sqrt{2ac}$. My textbook says the answer is $92ab^2\sqrt{2ac}$. Why doesnt ...
0
votes
1answer
23 views

Showing that $\mathrm{Rad}((0)) ≠ (0)$ implies $R^\times \subsetneq R[X]^\times$

Let $R$ be a commutative ring with $1$, and let $I ≤ R$ be an ideal. We call $\mathrm{Rad}(I) := \{r \in R: \exists n \in \mathbb{N}_0: r^n \in I\}$ the radical of $I$. I now want to show that if $\...
21
votes
7answers
4k views

How to show this formula to get a square root of a number in “just few seconds” is true?

I don't remember in which topic I found it but I know it was there. And I still have not find a proof of this nice approximation. Let $x$ be a non perfect square number. If $y$ is the closer ...
1
vote
1answer
66 views

Is this polynomial solvable by radicals?3

Suppose you have a field $\mathbb{F}$. Show that the polynomial $x^n-n\cdot1_{\mathbb{F}}\in \mathbb{F}[x]$, where $n\geq 2$ is solvable by radicals.
4
votes
5answers
462 views

What is the fastest method to find which of $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ and $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $ is bigger manually?

What is the fastest method to find which number is bigger manually? $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $
0
votes
2answers
95 views

Why we square while doing the proof of √2 is irrational? [closed]

When we prove that $\sqrt 2$ is irrational by the method of contradiction, we assume $\sqrt 2$ is a rational number: $\sqrt 2 = a/b$ Squaring both sides, $2 = a^2/b^2$. Here is my question: is ...
2
votes
2answers
30 views

How to pull out coefficient from radical in an integral

I am in an online Calculus 2 class, and before my professor gets back to me, I was wondering if you guys could help. I was reading through an example: How was 1/27 pulled out from the coefficient ...
7
votes
2answers
105 views

How to arrive at Ramanujan's nested radicals?

Ramanujan found that $\sqrt[3]{\cos\left(\frac {2\pi}{7}\right)}+\sqrt[3]{\cos\left(\frac {4\pi}{7}\right)}+\sqrt[3]{\cos\left(\frac {8\pi}{7}\right)}=\sqrt[3]{\frac {1}{2}\left(5-3\sqrt[3]{7}\right)}$...
3
votes
2answers
194 views

Evaluate $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+…}}}}}$ [duplicate]

$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+...}}}}}$ My Attempt: I tried to use the regular way. $A=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+...}}}}}$ $A^2=1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{...
2
votes
2answers
72 views

How do I solve this System of Equations?

How do I begin to solve this system? $$x^2=y+a$$$$y^2=z+a$$$$z^2=x+a$$ Do I take the square roots of $x,y$ and $z$? If so, we get $$x=\pm\sqrt{a+y}$$$$z=\pm\sqrt{a\pm\sqrt{a+y}}$$$$y=\pm\sqrt{a\pm\...
-1
votes
1answer
57 views

Calculate $\int \frac{x^{\:}}{\sqrt{x^4+3}}\ dx$ [closed]

How to calculate $$\int \frac{x^{\:}}{\sqrt{x^4+3}}\ dx$$
0
votes
1answer
35 views

Radical of a direct sum of Lie algebra

If we take $L$ a finite dimensional Lie algebra on $\mathbb{R}$, $A$ a sub-abgebra and $I$ an ideal of $L$ such that $L=A \oplus I$ as vector spaces. We have that $rad(L)=rad(A) \oplus rad(I)$ as ...
50
votes
9answers
6k views

Square root confusion: Why am I getting an answer if it doesn't work?

Alright, so I have $\sqrt{x-15} = 3-\sqrt{x}$. I first square both sides to get $x-15 = (3-\sqrt{x})(3-\sqrt{x})$ which simplifies to $x-15 = 9 -6\sqrt{x} + x$. I solved for $x$ and got $x = 16$, ...
4
votes
2answers
164 views

How can I solve this hard system of equations?

Solve the system below \begin{align} &\sqrt {3x} \left( 1+\frac {1}{x+y} \right) =2\\ &\sqrt {7y} \left( 1-\frac{1}{x+y} \right) =4\sqrt{2} \end{align} Frankly I am disappointed, ...
1
vote
1answer
50 views

The identity $ \sqrt[n]{z}\sqrt[n]{w} = \sqrt[n]{zw}$ for complex numbers

In the general case, when $z$ and $w$ are two complex numbers, we have that $ (1) \sqrt[n]{z}\sqrt[n]{w} \neq \sqrt[n]{zw}$ For example, $\sqrt{-1}\sqrt{-1} \neq \sqrt{-1.-1} = 1$. However, there ...
0
votes
2answers
37 views

Locate the following numbers on the number axis. [closed]

a) $\sqrt {2}$ b) $3\sqrt {2}$ c) $\sqrt {8}$ How can I locate properly?
1
vote
0answers
58 views

How would you denest this radical using Ramanujan's Cubic Identity?

The identity states that given the cubic $y=x^3+ax^2+bx+c$, you have this equation: $\sqrt[3]{u+x_1}+\sqrt[3]{u+x_2}+\sqrt[3]{u+x_3}=\sqrt[3]{w+3\sqrt[6]{d}}$ where $$u=\frac {ab-9c+\sqrt{d}}{2(a^2-3b)...
4
votes
0answers
46 views

Computing cube roots using three number means

I've asked a question some time ago about Computing square roots with arithmetic-harmonic mean but it turned out that this method is exactly the same as Newton's method (or Babylonian method) for ...
0
votes
3answers
51 views

Why is $\sqrt{x/x^{-1}}$ OR $\sqrt{x/{1/x}}$ = $\lvert x\rvert$ and not just x

I have this task: Find equal expression to square root of fraction of x and its inverted value (this is translated from my mother tongue so I'm sorry if I've used incorrect terms). Anyway the starting ...