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. T

    MHB Root Calculations: Easiest Way to Solve?

    Im just wondering what is the easiest way to deal with calculations where roots are involved? For example how do you solve this one? \frac{\frac{1}{2}}{1-\frac{\sqrt{2}}{2}}Thank you for replies!
  2. T

    MHB Finding Primitive Roots Modulo 169: Is There a Smarter Way?

    How can I find a primitive root modulo 169? I found the primitive roots mod 13 by testing 2, and then concluding that any 2^k with (k, 12)=1 would do. So that gave me 2, 6, 7 and 11. But modulo 13 I have no idea how to start.. I’m sure there’s a smarter way than trying 2^the orders that divide...
  3. Petrus

    MHB When there is a double root for the eigenvalue, how many eigenvectors?

    Hello MHB, I got one question. If I want to find basis ker and it got double root in eigenvalue but in that eigenvalue i find one eigenvector(/basis) what kind of decission can I make? Is it that if a eigenvalue got double root Then it Will ALWAYS have Two eigenvector(/basis)? Regards, |\pi\rangle
  4. Sudharaka

    MHB Jordan Normal Form and Root Spaces

    Hi everyone, :) Here's a question that I encountered recently. I would appreciate if you could go through my solution and let me know if you see any mistakes or have any comments. Question: Given a linear transformation \(f:\,\mathbb{C}\rightarrow \mathbb{C}\) with matrix...
  5. anemone

    MHB Which has a larger root?

    Which is larger, the real root of x + x2 + ... + x8 = 8 - 10x9, or the real root of x + x2 + ... + x10 = 8 - 10x11?
  6. F

    Series Convergence: Is the Root Test Always Reliable?

    Homework Statement determine whether the series (1-1/n^(1/3))^n converge or diverge Homework Equations all the testing procedure The Attempt at a Solution So I did the root test first, but the limit on the inside is 1. I then tried the ratio test but then when I tried taking the...
  7. K

    If p is prime, then its square root is irrational

    Homework Statement Im trying to prove that if p is prime, then its square root is irrational. The Attempt at a Solution Is a proof by contradiction a good way to do this? All i can think of is suppose p is prime and √p is a/b, p= (a^2)/ (b^2) Is there any property i can...
  8. Sudharaka

    MHB Proving Root Space Invariancy of Linear Transformation

    Hi everyone, :) Here's a question that I don't quite understand. What I don't understand here is what is meant by root space in the context of a linear transformation. Can somebody please explain this to me or direct me to a link where it's explained?
  9. N

    Integral: square root of sum of trig polynomials

    Hi, I am trying to make progress on the following integral I = \int_0^{2\pi} \sqrt{(1+\sum_{n=1}^N \alpha_n e^{-inx})(1+\sum_{n=1}^N \alpha_n^* e^{inx})} \ dx where * denotes complex conjugate and the Fourier coefficients \alpha_n are constant complex coefficients, and unspecified...
  10. twoski

    Newton's Method - Cube Root Of 5

    Homework Statement Use Newtons method to compute the cube root of 5. Do the first 10 iterations. x_{(0)}=1 determine the fixed points of the iteration and determine whether they are repelling/attracting. if attracting, then determine if the convergence is linear or quadratic. draw the...
  11. A

    MHB Square Root Rules for Fractions: x∈[3,∞)

    \sqrt{\frac{x-3}{x}} = \frac{\sqrt{x-3}}{\sqrt{x}} that is not true for all x, it is true for x\in [3,\infty) I want to teach my students that the exponents distribute over fractions unless we have a case like that square root or any even root. what do you think ?
  12. S

    Principal root of a complex number

    Homework Statement I am doing a problem of a contour integral where the f(z) is z1/2. I can do most of it, but it asks specifically for the principal root. I have been having troubles finding definitively what the principal root is. Anyplace it appears online it is vague, my book doesn't...
  13. P

    Root 2 irrationality proof (geometric)

    I was looking over this proof and I have some questions: http://jeremykun.com/2011/08/14/the-square-root-of-2-is-irrational-geometric-proof/ Second paragraph, what does "swinging a b-leg to the hypotunese" mean? Also, where did the arc come from, I really don't understand also, the last part...
  14. C

    MHB Sum series- Prove the equality of ratio and root.

    I found this on the internet, but did not find the proof. Curious to me is that the the ratio and root test have the same conditions. How can i basically prove this equality? \frac{a_{n+1}}{a_{n}} = \sqrt[n]{a_{n}} Thank you!
  15. K

    How to calculate Derivative of sin sq. root x by definition?

    Homework Statement Evaluate derivative of (sin sq. root x) w.r.t x? Homework Equations Limit Δx--> 0 (sin√(x+Δx) - sin(√x)) / Δx The Attempt at a Solution i couldn't operate it from here... Δy = (2cos((√x+Δx) + (√x)) . sin((√x+Δx) - (√x)) / Δx...?
  16. MathematicalPhysicist

    Roots and root vectors of sp(4,\mathbb{R})

    I found that the cartan subalgebra of ##sp(4,\mathbb{R})## is the algebra with diagonal matrices in ##sp(4,\mathbb{R})##. Now to find out the roots I need to compute: ##[H,X]=\alpha(H) X## For every ##H## in the above Cartan sublagebra, for some ##X \in sp(4,\mathbb{R})## Now, I know that...
  17. anemone

    MHB Prove that A Real Root Exists in [-1, 1]

    Given f(x)=5tx^4+sx^3+3rx^2+qx+p for $f(x)\in R$. If $r+t=-p$, prove that there is a real root for $f(x)=0$ in $[-1,1]$.
  18. S

    What to do when Wolfram doesn't give answer?

    I'm trying to compute following integral (Wolfram doesn't give answer): \int\sqrt{E-Bk^{2}\frac{cos^{2}(kr)}{sin^{2}(kr)}-k\frac{cos(kr)}{sin(kr)}\sqrt{D+Fk^{2}\frac{cos^{2}(kr)}{sin^{2}(kr)}}-\frac{Ak^{2}}{sin^{2}(kr)}}dr where A,B,C,D,E,F,k are constants. Substitution t=sin(kr) leads to...
  19. S

    Rationalizing the denominator involving more than one square root

    Here's my problems: How might you "rationalize the denominator" if the expression is \frac{1}{2+7√2+5√3} or \frac{1}{\sqrt[3]{2}+1}? I know that in typical problems where we rationalize the denominator, we simply have to multiply the denominator and numerator by the conjugate of the denominator...
  20. E

    MHB What is the real root of x^5+5x^3+5x-1?

    Find the real root of x^5+5x^3+5x-1
  21. M

    How do you rationalize a demoninator if the denominator is a cube root

    Hi, I know how to rationalize a denominator when it is a square root monomial or a square root binomial (through conjugation). For example, for a square-root monomial: 5/√25 = [5(√25)]/ [(√25)(√25)] = [5(√25)]/25 = (√25)/5 = 1 or -1and, for a square-root binomial: 5/(5 + √25) = 5(5 -...
  22. Fernando Revilla

    MHB Root or Ratio Test: Interval of Convergence

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  23. R

    An Integral With A Square Root In The Denominator

    How would you integrate it? \int \frac{d \varphi}{\sqrt{1 + \frac{a^2 b^2 \sin^2 \alpha}{(a \sin \varphi + b \sin (\alpha - \varphi))^2}}} I know that solving it numerically would probably be easier, but I would prefer a closed form solution in this case.
  24. Y

    MHB How can I solve this integral involving cube roots?

    Hello I am working on this integral \[\int \frac{\sqrt[3]{x}}{(\sqrt[3]{x}+1)^{5}}\]I have tried using a substitution, I did: \[u=\sqrt[3]{x}+1\] and I got that the integral becomes: \[3\cdot \int \frac{(u-1)(u-1^{2})}{u^{5}}du\]I moved on from there, got a result, however it was not...
  25. A

    MHB Program for Approximating nth root of a Number

    I have a question for a programming exercise I'm working on for C. The problem is to "Write a program that uses Newton's method to approximate the nth root of a number to six decimal places." The problem also said to terminate after 100 trials if it failed to converge. Q1. What does "converge"...
  26. Nugso

    Rational Root Theorem Homework: Solving x4 - 4(x3) + 3(x2) -2x +1 = 0

    Homework Statement x4 - 4(x3) + 3(x2) -2x +1 = 0 Homework Equations Rational Root Theorem, q/p The Attempt at a Solution Hello everyone. Today, I've learned the rational root theorem( it's a bit late, isn't it? :( ) and thus wanted to see how it works. According to the...
  27. A

    Solving for "a" in Square Root Equation

    Homework Statement Get the value of a if \sqrt{6-\sqrt{a}}+\sqrt{6+\sqrt{a}}=\sqrt{14} The Attempt at a Solution nothing succesfull Feel free to move this thread,..I actually place it here to tap more brains
  28. B

    Root Mean Square Or Standard Deviation

    Hello, I've been trying to find online where I could calculate my grade based off a college curve. So the average grade on the test was a 63. The RMS is 16. I got a 86. So what will my grade curve to? This is out of 50 people. Also, the professor curves to a 70 (I think).
  29. U

    Prove that this equation has at least one real root

    Homework Statement Let f:R→R be a continuous and differentiable function, then prove that the equation f'(x)+λf(x)=0 has at least one real root between any pair of roots of f(x)=0, λ being a real number Homework Equations The Attempt at a Solution All that I know from Rolle's Theorem is...
  30. K

    Using Rolle's theorem to prove at most one root

    Homework Statement show that the equation x^3-15x+C=0 has at most one root on the interval [-2,2] Homework Equations The Attempt at a Solution I know I need to use Rolle's theorem but I'm not sure how to find the answer. Thanks.
  31. V

    Formula & Conversion with a square root

    I'm comparing the shear formula for a beam in english and metric. But it seems the formula or result don't match. In English, the formula is Vc=2*b*d*sqrt(Fc) Given b=11.81102 inches d=18.11024 inches fc=4000 psi Vc=2*b*d*sqrt(Fc)=27056 lbs Now converting the units in metric...
  32. pratikaman

    Root difference inequality

    How many integers satisfy {√n-√(23×24)}^2<1 I was able to solved this by trial and error method , but i want to know systematic step-wise solution.
  33. M

    What Is The Root of 5? Irrational

    I wonder how do I find the root of 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, and so on? And why the circle and the curve looks so smooth in the computer graphic software such as AutoCAD, Adobe Illustrator, etc., if the root is can not be found? It should be looks rough. Thank you
  34. J

    Does an algebraic function always ramify at a multiple root?

    Hi, Consider the algebraic function w(z) given by the expression f(z,w)=a_0(z)+a_1(z)w+a_2(z)w^2+\cdots+a_n(z)w^n=0 where f(z,w)is irreducible over the rationals, and the coefficients, a_i(z), polynomials with rational coefficients . Let z_s be a point such that f(z_s,w)=0 has roots with...
  35. Petrus

    MHB Integrate Sine and Square root Composite Function

    Hello MHB, I got stuck on integrate this function \int \frac{\sin^3(\sqrt{x})}{\sqrt{x}}dx my first thinking was rewrite it as \int \frac{\sin^2(\sqrt{x})\sin(\sqrt{x})}{\sqrt{x}}dx then use the identity \cos^2(x)+\sin^2(x)=1 \ \therefore \sin^2x=1- \cos^2(x) \int...
  36. S

    Deriving the Cube Root Formula with Newton-Rhapson's Method

    Homework Statement Derive cube root formula using Newton-Rhapson's method. x - y^3 = 0. Homework Equations xn + 1 = xn - f(xn)/f'(xn) The Attempt at a Solution I know that the solution is (2y + (x/y^2))/3 I tried using implicit differentiation and stuff but I can't get this out...
  37. P

    MHB Can a Primitive Root of p Also Be a Primitive Root of p^2?

    Show that if $x$ is a primitive root of p, and $x^{p-1}$ is not congruent to 1 mod$p^2$, then x is a primitive root of $p^2$
  38. Barioth

    MHB Two Integrals with Square Root

    Hi everyone I have these 2 integrate that I can't solve, I have tried them with mathematica and wolfram, but they can't find an answer, maybe someone have an idea on how I could tackle these 2 bad boy! The first one is \int{ \sqrt{ \frac{1+( \frac{1}{10}+ \frac{s}{25})^2}{ \frac {s}{10}+...
  39. I

    Root mean square velocity question

    If at 120 K the Vrms is v , then at 480 K it will be a. 1 b. 2 c. 3 d. 4 Which formula is to be used here? Plz solve this.
  40. O

    My Taylor Square Root C Program doesn't like me

    Homework Statement 4. Implement a simple method to find the square root of a double precision floating point number x. A simple method is to consider the error produced by a “guess” y of the solution. Square the value y and compare with the value x. If y is correct, the error e=|y2-x| where ||...
  41. F

    Control Systems - Root Locus

    Homework Statement Given the transfer function: R(s)=\frac{K}{s(s+2)(s^2+4s+5)} Find the following (needed to sketch the root locus: a) Number of branches b) Symmetry c) Starting and ending points d) behavior at infinity (asymptotes) e) Real axis breakaway and break-in points. f)...
  42. Albert1

    MHB What is the Sum of Two Sixth Powers?

    $\sqrt {x}+\sqrt {y}=35$ $\sqrt [3]{x}+\sqrt[3] {y}=13$ find x+y
  43. G

    Is Taking the Principal Square Root Always Necessary?

    I feel kind of ridiculous making this post, but here we go: What would be the correct answer to this question; Choose all the number sets (natural, integer, rational, or irrational were the only options given) that -√81 belongs to, and show how you found your answer. What I said was this...
  44. P

    MHB Is There a Strategy for Finding Primitive Roots of 2p^n If There is One for p^n?

    let p be an odd prime. Show that if there is a primitive root of p^n, then there is a primitive root of 2p^n. Strategy?
  45. S

    Square Root of Positive Operator

    Homework Statement Suppose U = T^2 + \alpha T + \beta I is a positive operator on a real inner product space V with \alpha^2 < 4 \beta . Find the square root operator S of U.Homework Equations The Attempt at a Solution Isn't this just the operator S \in L(V) such that S e_k = \sqrt{...
  46. F

    Find the derivative and critical numbers of a cubed root function

    1. Find the intervals of increase and decrease 2. C(x)=x^{1/3}(x+4) 3. C(x)=x^{4/3}+4x^{1/3}; C'(x)=\frac{4}{3}x^{1/3}+\frac{4}{3}x^{-2/3}=\frac{4x^{1/3}}{3}+\frac{4}{3x^{2/3}}=\frac{x^{2/3}}{x^{2/3}}*\frac{4x^{1/3}}{3}+\frac{4}{3x^{2/3}}=\frac{4x+4}{3x^{2/3}} I am wondering...
  47. B

    Question about root test for series

    Homework Statement Can you if given a sigma notation for an infinite series Ʃ 2^n/(4^n+1) rewrite as, Ʃ 2^n/(4^n+1^n) then doing the root test can you lim n→∞ n^√abs((2/(4+1))^n) which equals 2/5, and 2/5<1, therefore can i conclude that the series above converges? Sorry...
  48. M

    How Do You Find The Exact Value Of Square Root of 3, 5, 7, 11?

    Is there any method to find the exact value of the square root of 3,5,7,11,13,14,15,17,18, etc.? Thank you
  49. C

    Integral of 1 - 2*sinx in square root?

    Hello PF members, I want to solve this integral but I cannot find a method. \int\sqrt{1 - 2sin(x)}dx for 0 < x < ∏/6 Or more generally \int\sqrt{a - bsin(x)}dx for a > b How can I solve this? Thanks in advance
  50. S

    Solving inequalities algebraically, when root is 0

    Homework Statement Solve each inequality without graphing the corresponding function. State the solution algebraically and graph on a number line: x/x2-9≤0 so i factor out the denominator and get (x+3)(x-3) the root here is zero, but for some reason in the chart (for rational/reciprocal...
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