What is Root: Definition and 941 Discussions

In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They most often lie below the surface of the soil, but roots can also be aerial or aerating, that is, growing up above the ground or especially above water.

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  1. nukeman

    F of G function question - Square root inside a square root?

    Homework Statement The image has the question I don't quite understand! Homework Equations The Attempt at a Solution I understand how to get √(√2 - x) but I don't get how they end up with: 4√2 - x ?
  2. D

    When the Rational Root Theorem Fails

    On a math test, one of the questions was to solve -\sqrt{7-x}=-\frac{x^2}{2}+12x-10. I solved graphically with a calculator, but later tried to solve algebraically, when I had more time. The equation is equivalent (with extraneous solutions) to x^4 + 48x^3 +536x^2 -956x + 372=0. This quartic has...
  3. B

    Limit of K-Roots in Pascal's Triangle

    Me and a friend was contemplating the ratio of even to odd number in Pascals triangle. After some thought we arrived at a ratio like this when looking at a triangle of 2k-1 rows. \frac{2^{2k-3}+2^{k-2}}{3^{k-1}} - 1 As expected, this ratio grows exponentially. (I.e. almost all numbers in...
  4. R

    Square root within a square root

    Hoping someone can push me in the right direction with this one. Plume snookered. It's to simplify: 2√3(3+√3) Guessing first calculate (a^2 - b^2*c) in the square, though the 2 is throwing this an I'm not sure how the answer is 6√3 + 6, an not 18√3 ?
  5. nukeman

    Finding the domain of a function (Sqr root)

    Homework Statement Here is a picture of the question: Homework Equations The Attempt at a Solution Here it what I did... factored it...so 1 / (x-3)(x+17) domain: (-infinity, -3)(-3,17)(17, +infinity) or rather (-3,17) I know that is not correct. I am really stuck on this one!
  6. M

    Partial fractions for a cubic root in the denominator of integrand

    Homework Statement \int\frac{1}{x\sqrt[3]{x+1}}dx (That's a cubic root in the denominator, by the way. Not an x cubed.) The Attempt at a Solution I thought possibly partial fractions, but I've never seen it done with a root in the denominator. Integration by parts was...
  7. S

    Calculating Nth Root of Vector: Non-Integer Solutions

    Could someone please explain how to calculate the nth root of a vector, where n is not an integer?
  8. L

    Prove Square Root of 15 is Irrational

    Homework Statement Prove Square Root of 15 is Irrational The Attempt at a Solution Here's what I have. I believe it's valid, but I want confirmation. As usual, for contradiction, assume 15.5=p/q, where p,q are coprime integers and q is non-zero. Thus, 15q2 = 5*3*q2 = p2...
  9. G

    Is this Proof of √3 Irrationality Flawed?

    Homework Statement Prove that there is no rational x such that x2=3 2. The attempt at a solution Suppose that there is a rational x=\frac{a}{b}=\sqrt{3} and that the fraction is fully simplified. (ie. a and b have no common factor) Then a2/b2=3 which means a2=b2.3 and it follows...
  10. L

    Finding the cube root of 1 using Euler's formula

    Homework Statement I have found this video where there is this problem: Find the cube root of 1 Homework Equations I have found this video: Where from 8:02 to the end she solves this problem.The Attempt at a Solution My question is why is she making such a big fuss about this? Is the...
  11. J

    Field Extensions and Root Fields

    I have a quick question. How does the following look? Proposition: Every extension of degree 2 is a root field. Proof: Let F be a field. Let p(x) ε F[x] and suppose p(x) has degree n. Then p(x) has n roots, say c1,c2,...,cn. Let E be the extension of F that contains the...
  12. F

    Write answer with only one root

    Homework Statement At the end I get the answer: sqrt(sqrt(3)+2) It can be show that it is ( sqrt(6)+sqrt(2) ) / 2 How do I show this?, I have tried, to attempt it my manipulating the expression, but I can't seem to manage it.
  13. F

    Graphing a cubed root function

    1. Graph the following function 2. \sqrt[3]{(x^{2}-1)^{2}} 3. I got the first derivative to be \frac{4x}{3\sqrt[3]{x^{2}-1}} but am having trouble with the second derivative to get the concavity. So far I have...
  14. O

    Question about reducing a square root.

    IDK if this should be in the precalc section, but I was wondering how to reduce \sqrt{(3+\sqrt{5})} / \sqrt{(3-\sqrt{5})} to (\sqrt{5} + 1) /(\sqrt{5} - 1)
  15. A

    Is the Square Root Function Bijective in all Branches of Mathematics?

    In what branches of mathematics is this proven.. I have never seen a proof, so I wonder if anyone can give me the basics of what is done to proove it or got a link to a proof.. Edit: By square root I mean the positive square root.
  16. C

    Proof about m/nth root of a prime.

    Lets take a prime number and raise it to m/n where m and n are coprime. x,y are coprime and I want to show that this is irrational. Proof: let's assume for the sake of contradiction that P^{\frac{m}{n}}=\frac{x}{y} P is prime and m,n,x,y are integers. no we take both sides to the...
  17. J

    Baby rudin, Limit of nth root of p

    Part (b) of theorem 3.20 is to prove that the limit as n approaches infinity of the nth root of p equals one(for p>0). The proof given in the text uses some inequality derived from the binomial theorem which seems to me to just come out of nowhere and provide a completely unintuitive proof...
  18. H

    MHB Manipulation of negative square root of a negative term/#

    Suppose I have to solve for y: x\leq 1 (x - 1)^{2} = y So I know that (x - 1) will always be 0 or a negative, therefore I must take the negative square root of (x - 1): -\sqrt{(x - 1)^{2}} = -\sqrt{y} Am I to understand that this is the same as: -1 \cdot \sqrt{(x - 1)^{2}} = -1 \cdot...
  19. caffeinemachine

    MHB Square root in Q(root 2) means its in Z[root 2]

    Let $a,b \in \mathbb{Z}$, and if $a+b\sqrt{2}$ has a square root in $\mathbb{Q}(\sqrt{2})$, then the square root is actually in $\mathbb{Z}[\sqrt{2}]$. Only one approach comes to my mind. Let $r_1, r_2 \in \mathbb{Q}$ such that $a+b\sqrt{2}=(r_1+r_2\sqrt{2})^2$. This gives $a=r_1^2+2r_2^2...
  20. R

    Existence of the Square Root Proof

    I was playing trying to work through a proof in Apostol's Calculus and can't quite understand a step noted. This is from chapter 3, theorem 1.35. Every nonnegative real number has a unique nonnegative square root. The part where you are establishing the set S as nonempty so you can use LUB it...
  21. O

    MHB Showing that Q(sqrt(p)) is in Q adjoined the pth root of unity

    i am having trouble showing that \mathbb{Q}(\sqrt{p*}) \subset \mathbb{Q}(\zeta_{p}) where p* = (-1)^{\frac{p-1}{2}}p . in other words, if p = 1 (mod 4) then p* = p and if p = 3 (mod 4) then p* = -p. i encountered this in the context of galois theory and i have no idea how to start. it seems...
  22. M

    Approximate solution for square root of sum of squares

    Homework Statement If X^2=Sqrt(x1^2+x2^2+x3^2+...)=> X? and vice versa If X=x1+x2+x3+...=> X^2? Homework Equations The Attempt at a Solution
  23. J

    Digit-by-digit calculation of square root

    Years back i learned the digit-by-digit calcualtion of square root like this : http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Digit-by-digit_calculation But i don't know why this works. Wikipedia gives kind of an explanation but i don't understand it. How does this method work?
  24. M

    Why Isn't x^2-1 Divisible by m?

    I have a question; Let m have a prime factor p \equiv 1 (mod 4). Then Euler function \varphi(m) is divisible by 4. Let x = r^{\varphi(m)}, then m|(x^4-1) and x^4-1=(x^2-1)(x^2+1). As gcd(x^2-1,x^2+1)|2, either x^2-1 or x^2+1 is divisible by m. My book says here because of the nature of a...
  25. N

    Integration involving a square root function.

    Homework Statement Integrate: sqrt(1/4 + t^2 + t^4)The Attempt at a Solution I'm really not sure on how to go about integrating this, it's actually integrate from -1 to 1, the solutions manual has a method I'm not familiar with. I thought of factorising it first, although doing that hasn't...
  26. D

    Integral with a root in the denominator

    Homework Statement I have been hung up on this integral: (μ /4∏ε) ∫ dz/(√s^2 + z^2) Homework Equations The Attempt at a Solution i have tried a couple of different u-substitutions, and none are getting me anywhere. I do not that partial fractions, or by parts would help...
  27. J

    About a root in the topic of automorphism and fixed field. HELP

    About a root in the topic of automorphism and fixed field. HELP! Homework Statement Let m be a positive integer. Let \xi = \exp (2pi/m) . Then \xi is a primitive m-th root of unity. (I.e., \xi is a solution of \Phi_{m}(X):=(X^m - 1)/(X-1) .) If \phi \in...
  28. W

    The root second notation for collider energies

    Every paper I read about cross-section measurements from colliders has a line saying (for example): ...positron-electron annihilations at \sqrt{s} = 40 GeV are studied... 1) What does this mean? I'm guessing it means that in the CM frame, the energy of each beam is 40 GeV. 2) Why use...
  29. T

    Numerical method: iteratation to converge on root

    Hello! I'm having trouble understanding the method/reasoning behind finding the root of an equation though iterative convergence. x2 - 4x + 1 = 0 x2 = + 4x - 1 x = 4 - 1/x I can understand that once we input a 'root' the equation will equal be equal on both sides. (Due to the remainder...
  30. R

    Line integral - confusion on squares and square root terms

    line integral -- confusion on squares and square root terms Homework Statement Do you see where they have sqrt(16 sin^2t etc = 5? How do they get that, the answer should be 7, the square root of 16 is 4, sin^2 + cos^2 is 1 and the square root of 9 is 3, 3 + 4 = 7. It's like they're...
  31. H

    How to integrate sqrt((ax+b)/x) dx (square root of linear fractional function)

    Hi, simple question, but difficult to find an answer for me How to integrate sqrt((ax+b)/x) dx ? a,b constants and x variable if it matters, I would be happy if you could solve it just for both a,b >0 Thanks
  32. P

    MATLAB: Finding the 5th Root using Newton's Method

    Homework Statement The solution of the nonlinear equation x^5-P=0 gives the fifth root of the number P. A numerical solution of the equation can be calculated with Newton’s method. The solution process starts by choosing a value x1 as a first estimate of the solution. Using this value, a...
  33. B

    A close approximation for square root of 2.

    By chance I stumbled on this "almost" equality: \frac{1}{5}(1/2+2/3+3/4+4/5+5/6+6/7+7/8+8/9+9/10) ≈ √2 - 7.2×10^{-6} I'm just wondering, are these funny coincidences simply, well, coincidences :biggrin: or is there some kind of explanation? I've see a ton of other funny stuff like...
  34. T

    Square root of 3 is irrational

    I am trying to prove sqrt(3) is irrational. I figured I would do it the same way that sqrt(2) is irrational is proved: Assume sqrt(2)=p/q You square both sides. and you get p^2 is even, therefore p is even. Also q^2 is shown to be even along with q. This leads to a contradiction. However...
  35. D

    Looks same as distinct root solution

    ay'' + by' + cy = 0 1) for 2 distinct roots,the solution should be y = Ae^m1x + Be^m2x 2) for Real and equal roots should be y = e^mx(A + Bx) that is it. So if same roots as the 2nd one,one solution is y1(x)=Ae^mx, what about the second solution? there is a second solution given...
  36. K

    How do we know the degree of a root which is also a point of inflexion?

    Hey guys, I was looking at an exam I did last year and tried to solve a question, which at the time I couldn't do. Unfortunately I'm running into the same problem I had during the exam, so hear me out on this one Question: The graph below has equation y =ax(x-b)(x+c)^d. Write down the values...
  37. L

    How to Find the Root of f(x) = x^3 - x - 1 for Homework?

    Homework Statement How does one find the root of f(x) = x^3 - x - 1 ? Quadratic Equation only works on power of 2. I can't factor out an x to get a first term of x^2 because then Quadratic equation still won't work because the middle and last term would be messed up, I think. What...
  38. L

    Can you find the square root of n using only +, -, :, x ?

    Is it possible to find √n using only the times table and the 4 operations?
  39. L

    How do you find the cubic root of n without using log keys?

    what is the quickest way to find \sqrt[3]{n} [on a pocket calculator] whitout using any \sqrt{} or log key?
  40. Jalo

    Finding Homogeneous Solutions for a Second Order Differential Equation

    Homework Statement y''-4y'+4y=x*e2x I'm trying to find the homogenous solutions of this equation. I know there are two, but I can only find one. YH=> y''-4y'+4y=0 Homework Equations The Attempt at a Solution y''-4y'+4y=0 Using the characteristic function: a2-4a+4=a <=>...
  41. A

    Not a political discussion Which one is the root cause?

    America made a lot of enemies so they need a powerful military -or- America has a powerful military so they made a lot of enemies?
  42. I

    Find derivative of Square root (x + square root(x + x^(1/2))) Help

    Homework Statement Define f(x)=\sqrt{}(x + (\sqrt{}(x + \sqrt{}x) Determine where f is differentiable and compute the derivative Homework Equations f'(xo)= lim as x approaches xo (f(x) - f(xo))/(x - xo) The Attempt at a Solution By the definition, f(x) = \sqrt{}x does not have a...
  43. S

    Double Integration (Stuck at square root step) (Solution Included).

    Homework Statement The problem and solution are included. Homework Equations Double integration. The Attempt at a Solution Firstly, I'd like to mention that the additional ρ under the square root is there accidentally and that it should be outside of the square root such that it...
  44. S

    What did I do wrong here? (expressing root x as taylor series about a=4)

    Homework Statement Here is the question: I don't quite know what I did wrong. My method is below. Homework Equations The Attempt at a Solution f(x)=√x f'(x)=\frac{1}{2(x)^{1/2}} f''(x)=\frac{-1}{(2)(2)(x^{3/2}} a=4 f(a)=2 f'(a)=1/4...
  45. K

    Approximating Root of Equation with Newton's Method Using Maple

    Homework Statement Use Newton’s Method to approximate the indicated root of the equation to correct six decimal places. The root of 2.2x5 – 4.4x3 + 1.3x2-0.9x-4.0=0 in the interval [-2, -1] USE MAPLE. Homework Equations Newtons'method. The Attempt at a Solution I am...
  46. X

    Solving a Cubic Equation: How to Find the Real Root

    Homework Statement 2t^3-4t-5=0 Find t Homework Equations [-b-sqrt(b^2-4ac)]/2a The Attempt at a Solution The answer is sqrt(2/3). No idea how this came about. I tried plugging in the equation and got it wrong. :( Please help I've been struggling over this for 5 hours.
  47. C

    How can the endless square root problem be solved?

    I'm having a little bit of trouble figuring out how exactly to do this. Prove that \sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}}} = \frac{1\pm\sqrt{4n+1}}{2}. How exactly does one go about doing this? I mean, I understand it goes on infinitely, but doesn't that create an infinitely...
  48. karush

    MHB What is the simplified form of the limit as x approaches infinity?

    \lim_{x\to +\infty}\sqrt{x^2+3}-x sorry first of all how do you turn this string into latex I don't see the icon tool on the editor thnx
  49. J

    Qube root of 2, zero of second order polynomial

    How do you prove that there does not exist numbers a,b\in\mathbb{Q} such that 0 = a + b\sqrt[3]{2} + \sqrt[3]{2}^2
  50. S

    Existence of a root between 2 given points

    Homework Statement Show that there exists one root int (0,2) of the following function: f(x)=(1-x^2)^2-√((1-x^2)*(1-1/2*x^2)) Homework Equations The Attempt at a Solution I first found: f(0)=0 and f(2)=7.268 But, i don't know what to do now. I'm not sure if it has...
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