Root Definition and 944 Threads

Quick question: In the proof that the square root of 2 is irrational, when we are arguing by contradiction, why are we allowed to assume that ##\displaystyle \frac{p}{q}## is in lowest terms? What if we assumed that they weren't in lowest terms, or what if we assumed that ##\operatorname{gcd}...

Homework Statement The problem relates to a proof of a previous statement, so I shall present it first: "Suppose P is a self-adjoint operator on an inner product space V and ##\langle P(u),u \rangle## ≥ 0 for every u ∈ V, prove P=T2 for some self-adjoint operator T. Because P is self-adjoint...

Given two positive numbers a and b, we define the root mean square as follows: R. M. S. = sqrt{(a^2 + b^2)/2} The arithmetic mean is given by (a + b)/2. Given a = 1 and b = 2, which is larger, A. M. or R. M. S. ? A. M. = sqrt{1•2} A. M. = sqrt{2} R. M. S. = sqrt{(1^2 + 2^2)/2} R. M. S. =...

Hi guys. I was wondering something. In my math class, we were analyzing how strong the data was, and there was an r and r^2 value. I know the significance of r, but what's the point of knowing the square of the r value? Also, what's the use of square root? Like where does it help? I saw it one...

Homework Statement How to integrate ## \frac{dx}{dt}=\sqrt{\frac{k}{x}-1}## AND ## \frac{dx}{dt}=\sqrt{\frac{k}{x}+1}## k a constant here. I'm unsure what substitution to do. Many thanks in advance. Homework EquationsThe Attempt at a Solution I can't really get started as I'm unsure...

]Homework Statement Solving the cubic equation x^3 + 6x = 20 by using formula gives (10+ sqrt(108))^1/3 - (-10 + sqrt(108))^1/3 How do you show that this comes out exactly 2? No calculators allowed.Homework EquationsThe Attempt at a Solution Tried cubing the expression and tried using the...

EDIT ... ... SOLVED ...Can anyone help me with displaying the cube root of 2 as shown in the example below taken from Dummit and Foote, Section 13.4 ...https://www.physicsforums.com/attachments/6605Help will be appreciated, Peter*** EDIT *** Just found what I think is the solution ... it's...

I know that x^2 = 4 yields two answers: x = -2 or x = 2. I also know that x^3 = 8 yields x = 2. Question: Why does the square root yield both a positive and negative answer whereas the cube root yields a positive answer?

Homework Statement For an exam question i need to be able to sketch the root locus of a system, for example the following: g(S) = 200(S+3) / ((S+2)(S+4)(S+6)(S+8)(S+10) The Attempt at a Solution So i counted number of poles and zeroes and calculated no. of asymtodes: p-z = 4 and calculated the...

When we find solution set of an equation inside a square root why we should assume that inside of square root should be equal to or greater than zero? For example ##\sqrt{5x-4}##. How can I use here equal to or greater than zero symbol? Thank you.

Let cbrt = cube rootcbrt{3} x cbrt{3} = (3)^(1/3) * (3)^(1/3) 3^(1/6) ir sixth root {3} Correct?

Hello everyone I am trying to write code in ROOT.I want to plot generalized hypergeometric function pFq with p=0 and q=3 i.e I want to plot 0F3(;4/3,5/3,2;x) as a function of x using TF1 class.I am not getting how to plot this function in ROOT.Kindly help me out. Thanks in Advance

Why do we get two answers when taking the square root? For example, let a = any positive number sqrt{a} = - a and a. Why is this the case? What about 0? Can we say sqrt{0} = - 0 and 0?

How does the value of ##\displaystyle \sqrt[a]{-1}## vary as ##a## varies as any real number? When is this value complex and when is it real? For example, we know that when a = 2 it is complex, but when a = 3 it is real. What about when ##a = \pi##, for example?

1. Homework Statement $$\frac{dy}{dx}=\sqrt[3]{\frac{y}{x}},~x>0$$ Why do i need the x>0, indeed my result is good for all x since it contains x2 2. Homework Equations $$\frac{dy}{dx}=f(x)~\rightarrow~dy=f(x)dx~\rightarrow~y=\int f(x)dx$$ 3. The Attempt at a Solution $$\int...

Hello,first time posting a thread not just here but generally so i'll try my best. So while i was in class we were learning about square roots,at first it seemed fairly easy,but when i asked my math teacher how do we find them more easily, he smiled and talled me:"The problem is,you just...

Hi everyone, What is the solution set of the equation: sqrt{x+2}= x-4 I got 2 and 7. Is it correct or is it just 7. If so why? Thanks:)

I'm not sure which category to post this question under :) I'm not sure if any of you are familiar with "Greek Ladders" I have these two formulas: ${x}_{n+1}={x}_{n}+{y}_{n}$ ${y}_{n+1}={x}_{n+1}+{x}_{n}$ x y $\frac{y}{x}$ 1 1 1 2 3 1.5 5 7 ~1.4 12 17 ~1.4 29 41...

Is square root of delta function a delta function again? $$\int_{-\infty}^\infty f(x) \sqrt{\delta(x-a)} dx$$ How is this integral evaluated?

I have the expression ##e^{\frac{1}{2} \log|2x-1|}##. I am tempted to just say that this is equal to ##\sqrt{2x-1}## and be done with it. However, I am not sure how to justify this, since it seems that then the domains of the two functions would be different, since the latter would be all real...

Hello I am trying to solve this limit here: \[\lim_{x\rightarrow -\infty }\sqrt{x^{2}+3}+x\] I understand that it should be 0 since the power and square root cancel each other, while the power turned the minus into plus, and then when I add infinity I get 0. This is logic, I wish to know how...

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 let c be a primitive 16th root of unity. How many subfields M<Q(c) are there such that [M:Q] = 2 Homework EquationsThe Attempt at a Solution I think the only subfield M of Q(c) such that [M:Q] = 2 is Q(c^8). Then M = {a+b(c^8) such that a,b are elements of Q}. I'm thinking...

Why √[(-a).(-b)] can't be written as √(-a).√(-b) Is it only because complex number do not work for this statement. Just like here: √ab = √[(-a).(-b)] = √a√bi^2 = -√ab which is wrong. We can separate √(-4)(9) = √-36 = 6i , √4i.√9 =6i, but why can't we separate for two negative numbers inside...

Hey guys, Trying to design a spur gear but I am very confused as the root circle/dedendum ends up being greater than the base circle. What do I do in this case? The gear I'm trying to design has a 68.33mm pitch diameter, 60 teeth, the pressure angle a standard 20 degrees. What am I doing...

http://mathhelpboards.com/pre-algebra-algebra-2/find-value-squareroot-3-using-graph-drawing-suitable-straight-line-19973.html I guess I found a method to obtain the square root of any number using the above graph. $x^2-2x-3$ What I did to find the square root of 3 was replace $x^2$ with the...

Homework Statement So I need the find the minimal polynomial of the primitive 15th root of unity. Let's call this minimal polynomial m(x) Homework EquationsThe Attempt at a Solution I know that m(x) is an irreducible factor of x^15 - 1 and also that the degree of m(x) is equal to the Euler...

$\tiny{206.b.46}$ \begin{align*} \displaystyle S_{46}&=\sum_{k=1}^{\infty} \frac{2^k}{e^{k}-1 }\approx3.32569\\ % e^7 &=1+7+\frac{7^2}{2!} %+\frac{7^3}{3!}+\frac{7^4}{4!}+\cdots \\ %e^7 &=1+7+\frac{49}{2}+\frac{343}{6}+\frac{2401}{24}+\cdots \end{align*} $\textsf{root test}$...

Let $f:\mathbf R\to \mathbf R$ be a smooth map and $g:\mathbf R\to \mathbf R$ be defined as $g(x)=f(x^{1/3})$ for all $x\in \mathbf R$. Problem. Then $g$ is smooth if and only if $f^{(n)}(0)$ is $0$ whenever $n$ is not an integral multiple of $3$. One direction is easy. Assume $g$ is smooth...

I tried to calculate the cubic root by using the method that are exist in Numerical receipes 77 but I got no answer and I don't know my mistake . Also, I tried by using Cardino method but Also I couldn't success to get an answer. Can any read my codes and tell me where is my errors or provide me...

Homework Statement Show, using the axiom of completeness of ##\mathbb{R}##, that every positive real number has a unique n'th root that is a positive real number. Or in symbols: ##n \in \mathbb{N_0}, a \in \mathbb{R^{+}} \Rightarrow \exists! x \in \mathbb{R^{+}}: x^n = a## Homework...

I am using ROOT to calculate the Fourier transform of a digital signal. I can extract the individual parts of the transform, the magnitude and phase in the form of a 1D histogram. I am attempting to reconstruct the transforms from the phase and magnitude but cannot seem to figure it out. Any...

Homework Statement Find the value of the integral ## \int_0^\infty dx \frac{\sqrt{x}}{1+x^2} ## using calculus of residues! Homework EquationsThe Attempt at a Solution This is how I did it: ##\int_0^\infty dx \frac{\sqrt{x}}{1+x^2}=\frac 1 2 \int_{-\infty}^\infty dx \frac{\sqrt{|x|}}{1+x^2} ##...

Let's say there's an equation 0 = √x - √x I intend to make x the subject of the equation; however because it is a square root, there are numerous solutions; however can I just assume that 0= √x - -√x= 2√x Can I now just rearrange this equation to make x the subject? In other words is the...

I'm trying to decide if simplifying sqrt(y^6) requires use of the absolute value bars. For example, the rule "nth root(u^n) = abs(u) when n is even" can be used to simplify sqrt(y^6) as sqrt[(y^3)^2]=abs(y^3). However, the rules of rational exponents can also be used to simplify sqrt(y^6) as...

Homework Statement There is almost no helium gas in the earth’s atmosphere - indeed the price of He has increased in recent times due to worries about a limited supply. (Bad news for parties and for all the scientists who use liquid He as a coolant.) we know that the “escape velocity” required...

Using a graph of function $y=3-(x-1)^2$ which has got its negative & positive root s-0.8 and 2.7 respectively, Find an approximate value for $\sqrt{3}$. Any suggestions on how to begin? Should I be using the quadratic formula here? Many Thanks :)

The main problem is http://mathhelpboards.com/pre-algebra-algebra-2/find-length-dc-19355.html#post88492 In this question $15 = \dfrac{\left((x+3)+(2x-3)\right)h}{2}=\frac12 ((x+3)+(2x-3))\times((2x-3) -(x+3))=\frac12((2x-3)^2-(x+3)^2)=\frac12(3 x^2-18 x)$ So we get $30=3x^2-18x$ Now using...

Question 1: (a) Show that the complex number i is a root of the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 (b) Find the other roots of this equation Work: Well, I thought about factoring the equation into (x^2 + ...) (x^2+...) but I couldn't do it. Is there a method for that? Anyways the reason I...

Does someone know how I can make such a legend entry in ROOT? I have tried all the draw options ("L","F","E") but I am unable to get this result. It really looks like a mixture of LF, but LF draws me a rectangular box enclosed in black line. extra info: I am using two graphs, one is for the...

The problem $$ \lim_{x \rightarrow \infty} \left( \sqrt{x+1} - \sqrt{x} \right) $$ The attempt ## \left( \sqrt{x+1} - \sqrt{x} \right) = \frac{\left( \sqrt{x+1} - \sqrt{x} \right)\left( \sqrt{x+1} + \sqrt{x} \right) }{\left( \sqrt{x+1} + \sqrt{x} \right) } = \frac{x+1 - x }{\left(...

Suppose [K:F]=n, where K is a root field over F. Prove K is a root field over F of every irreducible polynomial of degree n in F[x] having a root in K. I don't believe my solution to this problem because I 'prove' the stronger statement: "K is a root field over F for every irreducible...

Homework Statement [/B] f(x) = 2x3+ax2+bx+10 When f(x)/(2x-1) the remainder is 12 When f(x)/(x+1) there is no remainder a) Find the value of a and b b) Show that f(x) = 0 has only one rootHomework Equations None The Attempt at a Solutiona) (2x-1)=0 x=1/2 f(1/2) = 12 =...

I have this expression: $$\sqrt{ 1 - \frac{16}{\sqrt{x^2 + 16}}}$$ And the textbook simplifies it to $$\frac{x}{\sqrt{x^2 + 16}}$$ But I'm not sure how it does this.

Prove that if \(p(x)=x^4+ax^2+b\) is irreducible in F[x], then \(F[x]/<p(x)>\) is the root field of p(x) over F. My Attempt: 1. Let F(c) = \(F[x]/<p(x)>\) where c is a root of p(x). Then F(c) is a degree 4 extension over F because c is the root of a 4th order irreducible polynomial in F[x]...

Root of (-2-3) ^2 = -5 ( because root of squared number is the number itself) but alsoo square of (-2-3) is 25 and its root is (+5) /(-5). Therefore what is the correct answer and reason . I think it is -5(google answer is Also -5) but I don't have any reason. Please help me

$(a-1)x^2-(a^2+2)x+(a^2+2a)=0----(1)\\ (b-1)x^2-(b^2+2)x+(b^2+2b)=0----(2) $ if $(1)$ and $(2)$ have one root in common , (here $a,b\in N$ ,$a\neq b,\,\, and \,\, a>1,b>1$) find value of : $\dfrac{a^a+b^b}{a^{-b}+b^{-a}}$

if for rational a,b,c $ax^3+bx+c=0$ one root is product of 2 roots then that root is rational

Given a polynomial ##f(x)##. Suppose there exists a value ##c## such that ##f(c)=f'(c)=0##, where ##f'## denotes the derivative of ##f##. Then ##f(x)=(x-c)^mh(x)##, where ##m## is an integer greater than 1 and ##h(x)## is a polynomial. Is it true? Could you prove it? Note: The converse is true...

Is this an abuse of Rolle's theorem? Rolle's theorem If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0. ##[x_1, x_1]##...