Radicals Definition and 103 Threads

Hi PF. I'm aware of a formula for solving (when possibile) nested square roots of the type ##\sqrt{a+\sqrt{b}}##. But is there any formula/strategy for solving higher degree nested radicals? For example, I cannot understand how one can solve... $$ \sqrt[5]{\frac{5 \sqrt{5}-11}{2^6}}=\frac{1}{4}...

Hello, as the title says, how do you treat singly occupied orbitals in radical molecules, when trying to identify the HOMO and LUMOs? In the majority of cases I stumbled upon, like O2, both the antibonding orbitals, pi*2px pi*2py, are singly occupied, so they would be considered the HOMO, with...

In general, algebraic equations cannot be solved in radicals (we know it, e.g., from the Galois theory). So how can we solve such equations? We can always solve them approximately on a computer, but that's not what I'm asking about. Is there an exact analytic method to solve at least some of...

There appears to be a simple isomorphism between continued fractions and nested radicals. Does anybody know more about this?

find limit $\displaystyle \lim_{x \to 2}\dfrac{\sqrt{6-x}{-2}}{\sqrt{3-x}-1}$ ok I presnume the first thing to do is mulitply by conjugate

Alright so in organic chemistry we learn that for radicals and carbocations, the tertiary radical is the most stable (of primary and secondary). Okay, so why when we are predicting the major product of the reaction, let's take Br for example, the Br is going to attach to the tertiary radical. If...

Homework Statement The current section I'm working on has to do with arc length of a curve and surface area. These all eventually end up with having to take the anti-derivative of a radical. At each instance, I get stuck by using u-substitution because when I take the derivative of ##u##, my...

In the thread https://www.physicsforums.com/threads/recursive-square-root-inside-square-root-problem.954655/ a sequence interpreted from the notation: ##\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}## was discussed. What sequence would you (fellow forum members) associate with...

I am seeing from my results that an oxidation process is taking place, the pollutants degradation correlates closely with the formation of radicals. However, in one case there are less radicals produced and there is no degradation... I would like to ask if on occasions there must be a...

I'm looking at the oxidation of I- ions by OH radicals to form iodine and finally triiodide. This is a well known method of dosimetry, however some consider it flawed because during disassociation of solution other radicals are produced such as H or O radicals. They could also potentially...

I know my last question on radicals' interaction with luminol didn't get much attention, perhaps it was too specific. Please could I ask a last question, I understand that when radicals are formed via bond breaking this is called the bond-dissociation energy, but is there an energy associated...

Prove the following identity ($n = 1,2,3,...$): \[\sqrt{n - \sqrt{n+\sqrt{n-\sqrt{n +...}}}} = \sqrt{(n-1)-\sqrt{(n-1)-\sqrt{(n-1)-...}}}\]

Homework Statement Let ##S## be a multiplicative subset of a commutative ring ##R## with identity. If ##I## is an ideal in ##R##, then ##S^{-1}(\mbox{ Rad } I) = \mbox{Rad}(S^{-1}I)##. Homework EquationsThe Attempt at a Solution If ##x \in S^{-1}(\mbox{ Rad } I)##, then ##x = \frac{r}{s}##...

Homework Statement Find the differential Homework Equations Chain rule : dy/du=dy/du*du/dx Product rule: f(x)g'(x) + g(x)f'(x) The Attempt at a Solution I have tried to move the radical to the top of the equation by making it into an exponent (x^2+1)^-1/2. I then used the product rule and the...

Hi everyone. This is my proof (?)of ramanujan's problem 525: http://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/q525.htm (link to problem) [![enter image description here][1]][1] $$ \sqrt{A^{1/3}-B^{1/3}}=\frac{(A*B/10)^{1/3}+(A \times B)^{1/3}-(A^2)^{1/3}}{3} \Leftrightarrow \\ 9...

Hey! :o We have that $E/F$ is an extension Kummer of degree $n$ and that $F$ contains a $n$-th unit root $\omega$ with $\text{ord} (\omega)=n$. I want to show that $E/F$ is an extension with radicals of order $n$. I have found the following theorem: Could we maybe use that theorem in...

I became curious about the following problem from a discussion in another thread: https://www.physicsforums.com/threads/showing-a-polynomial-is-not-solvable-by-radicals.895282/ After a bit of study I concluded that the meaning of the assertion below regarding some specific real number rl P has...

Homework Statement Show that the polynomial f(x)=x^5-3x^4+6x^3+18x^2-3 is NOT solvable by radicals Homework EquationsThe Attempt at a Solution I'm pretty sure that to prove that this polynomial is not solvable I am too show that it has exactly 3 roots. That means that it will have 3 roots and...

Homework Statement Show that the polynomial f(x) = x^5 - x^3 - 3x^2 + 3 is solvable by radicals where the coefficients of f are from the field of rational numbers. Homework EquationsThe Attempt at a Solution My strategy to solve this problem was to construct a splitting field and then see if...

Homework Statement lim as x tends to -∞ (x)^3/5 - (x)^1/5 Homework EquationsThe Attempt at a Solution The first thing I did was convert it into a radical so it becomes fifthroot√x^3 - fifthroot√x. Then I rationalized to get ( x^3-x)/(fifthrt√x^3+fifthroot√x) . I then divided the top by x^3...

Hi I'm trying to give myself a refresher in Leaving Cert maths and I'm running through some problems. Here's one which has me stumped (sorry I can't figure out how to show the actual symbols on the post, it's just showing as raw LaTEX when I try ) Combine terms and simplify the expression of -...

I am solving the equation ##\sqrt{x + 3} + 4 = \sqrt{8x + 1}##. I understand that , generally, to solve it, we have to eliminate the radicals by isolating a radical expression to one side and then squaring both sides of the equation. I end up obtaining two solutions: ##x = 6## and ##x =...

Homework Statement Homework Equations Vieta's formula, quadratic formula The Attempt at a Solution I did use Vieta's formula, and got a set of equations in terms of α, however the tricky bit is to deal with the cube root. Then I tried the quadratic formula, but the algebra becomes too...

Hi all, I am learning about magnetometers. Esp. those based on the Overhauser effect (https://en.wikipedia.org/wiki/Nuclear_Overhauser_effect). As I understand, they embed a free radical fluid that contains free electrons that gets polarized through a RF field. Do you have any idea what free...

This may seem a bit pedantic but I'm writing a report and have radicals in it and notice that some papers put the dot in different places, so my question is does it matter where you put the dot? thanks for any advice

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote (D&F) Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.2 Radicals and Affine Varieties ... ... I need help with an aspect...

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.2 Radicals and Affine Varieties ... ... I need help on an apparently simple...

A friend and I have been working on this problem for hours - how can the following expression be simplified analytically? It equals $\frac{1}{2},$ and we have tried the following to no avail: 1. Substitution of $x = \sqrt{5}$ 2. Substitution of $x = 2\sqrt{5}$ 3. Substitution of $x =...

I did this problem on paper but my calculator doesn't agree with the result. Can somebody tell me where I'm going wrong and how to do it right? (7-sqrt3a)(7+sqrt3a) = (7-sqrt3a)7+(7-sqrt3a)sqrt3a = (7•7-7sqrt3a)+(7sqrt3a-sqrt3asqrt3a) = 49-7sqrt3a+7sqrt3a-3a = 49+3a-14sqrt3a Even before I...

Prove that for any natural number $a$, there is an integer $k$ such that $(\sqrt{1982}+1)^a=\sqrt{k}+\sqrt{k+1981^a}$

Homework Statement Hello , I need to find the real number solutions for the following equation. \sqrt{a-x} + \sqrt{b-x} = \sqrt{a+b-2x} where b>a>0 Homework Equations equation is given above The Attempt at a Solution I squared both sides and and solved this. I got two solutions x=a and x=b...

I want to de-nest the following radical: (1) \sqrt{3+2\sqrt{2}} Into the general simplified form: (2) a+b\sqrt{2} Equating (1) with (2), (3) \sqrt{3+2\sqrt{2}} = a+b\sqrt{2} and squaring both sides: (4) 3+2\sqrt{2} = a^2 + 2b^2 + 2ab\sqrt{2} generates a system of two equations with two...

how is $\sqrt{6}-\sqrt{2}$ equal to $2\sqrt{2-\sqrt{3}}$ please explain. Thanks!

Homework Statement Find the limit of the following sequence: Homework Equations \lim_{n \rightarrow + \infty} \sqrt[4] {2n + 1} - \sqrt[4] {n + 1} The Attempt at a Solution I've tried multiplying the first radical by ## \frac{ \sqrt[4] {2n - 1} } { \sqrt[4] {2n - 1} } ## to make the radical...

I am trying to find a general expression for radicals, for example, √k=f(k) Does anyone know of any? Thanks!

Homework Statement Hey guys/gals I have need a clarification on one particular pre-algebra problem dealing with multiplying radicals. I thought I knew the steps to solve it (properties of radicals and distribution property, etc) but I am having trouble with this particular problem. (8√6...

I am reading R.Y. Sharp: Steps in Commutative Algebra, Chapter 3: Prime Ideals and Maximal Ideals. Exerise 3.47 on page 52 reads as follows: ===================================================== Let P be a prime ideal of the commutative ring R. Show that \sqrt (P^n) = P...

can you tell me what is the proper conjugate of the denominator. what is the rule on how to group this kind of denominator to get the conjugate.$\displaystyle \frac{2+\sqrt{3}+\sqrt{5}}{2+\sqrt{3}-\sqrt{5}}$ thanks!

:mad: I really hate these problems. (2+√7)/(3-√-11). What the heck? I start out by multiplying both sides with the conjugate again. This is where I am stuck lol. Can someone tell me what I am doing wrong while multiplying the conjugate? (3 - √-11) is the same as 3 - √11i correct? So I...

I have some problems I am stuck on. The goal here is to simplify radicals in the denominator. I understand that when there is a binomial in the denominator, you need to multiply both sides by the conjugate. For some reason though, I seem to be having trouble doing that or am making a mistake...

can you tell me if there's a necessity to use the definition: $\displaystyle \sqrt{x^2}=|x|$ to this, $\displaystyle \sqrt{(x+y)^2}$ if yes, why? if not why? and how it is different to $\displaystyle \left(\sqrt{(x+y)}\right)^2$ thanks!

The aims of this tutorial are threefold: (1) By assuming two values of the Cosine function - which will be proven later on - we develop a large number of multiply-nested radicals to express \cos(\pi/2^kn),\, \sin(\pi/2^kn), \, and \tan(\pi/2^kn) in closed form, for ever smaller arguments (k...

Homework Statement limx->4 (sqrt(5-x)-1) / (2-sqrt(x)) NOT ALLOWED TO USE L'HOSPITALS. Homework Equations The Attempt at a Solution I tried using conjugate of both top and bottom and I couldn't get it to work, but maybe I've done it wrong?.. I'm not allowed to use...

\lim_{x \to a} \frac{ x^2 - a^2}{\sqrt(x) - \sqrt(a)} I've tried to solve this standard, but I either end up with 0 in the denominator, or I end up with 0/0. Any hints on what to do with this next? Thanks

I know how the very basics but then I get given a question like this. ##\sqrt{9-3x}## and I think I can divide both by 3. ##3\sqrt{3-x}## and so ##x=3## Then I get ##\sqrt{4x+12}## and again I can take 4 from each ##4\sqrt{x+3}## and so ##x=-3## Is this the correct way to be solving these...

Hi, can anyone please explain what selectivity of free radicals is? For example, there is an MCAT question that I recently came across and that can't seem to make sense of the answer. Which of the following halogens will give the greatest percent yield of tertiary alkyl halide when reacted...

Okay so I'm in Calculus 1 and we are working on derivatives. I understand it all but I have been having some trouble with some basic math skills that I cannot remember from high school and I can't seem to find a good tutorial anywhere online. I am having problems with multiplying fractional...

When calculating the limit of the function f(x) = (x^2 + 3)/ sqrt(2x^4 + 5) as x→∞, is it correct to square the top and then place the resulting polynomial under a square root (i.e. sqrt(x^2 + 3)^2)? Then you can rewrite the problem as the square root of the limit as x→∞ of the resulting...

Homework Statement for the following integrals, am I allowed to break them up like so: 1. ∫(1)/(sqrt(16-9x²)³) dx = ∫(1)/(√16)³ · ∫(1)/(√-9x²)³ dx 2. ∫(x²)/(sqrt(x²-9)) dx = ∫(x²)/(√x²) · ∫(x²)/(√-9) dx 3. ∫(1)/(x²(sqrt(a²+x²))) dx = ∫(1)/(x²) · ∫(1)/(√a²) · ∫(1)/(√x²) dx...

Homework Statement Evaluate lim x →0 √(x+1) - √(2x+1) ----------------- √(3x+4) - √(2x+4) Homework Equations The Attempt at a Solution First I rationalized the numerator by multiplying everything by √(x+1) + √(2x+1) / √(x+1)...