Root Definition and 944 Threads

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. M

    How should I show that this root is given approximately by this?

    Proof: Consider the equation of ## a^2-x^2=\epsilon\sinh x ## for ## 0<\epsilon<<1 ##. Let ## \epsilon=0 ##. Then the unperturbed equation is ## a^2-x^2=0 ##. This gives ## a^2-x^2=0\implies (a+x)(a-x)=0\implies x=\pm a ## with the root ## x=x_{0} ## such that ## x_{0}(a)=a ## because ## x\geq...
  2. S

    Find interval and root of polynomial with absolute error less than 1/8

    By IVT and trial and error, I get the interval to be ##(-\frac{1}{2},-\frac{1}{4})## I don't know how to do the next part. Let the actual root of the polynomial be ##x_{0}## and the approximate value is ##p##, we have ##|p-x_{0}|<\frac{1}{8}## I am not sure how to continue. Thanks
  3. L

    Can square roots be simplified without a calculator?

    I can't find a short solution without using calculator.
  4. J

    I Noise Proportional to Square Root of Illumination: Need Formula Help

    Many people have said that the noise that affects laser light is proportional to the square root of the illumination. But I can't find the formula. Can anyone help?
  5. chwala

    Find the root of the given equation in terms of ##u##

    Hmmmm was a nice one... took me some time to figure out ...seeking alternative ways ... My working; ##x=\dfrac{-2±\sqrt{4+4 \sinh^2 2u}}{2 \sinh 2u}## ##x=\dfrac{-2±2\sqrt{1+ \sinh^2 2u}}{2 \sinh 2u}## ##x=\dfrac{-1±\sqrt{1+ \sinh^2 2u}}{\sinh 2u}## ##x=\dfrac{-1+\sqrt{1+ \sinh^2 2u}}{\sinh...
  6. curiousPep

    Engineering Can I use root locus when the input is the negative feedback?

    I have used root locus before but my confusion now is that the input is the negative feedback. Usually when I have negative feedback I consider the the error between the input (ideal) signal and the observed signal. Also, in this case what is the tranfer function since u = -k*y, and what does...
  7. N

    I Calculating the root of a number by hand

    Hi, is it possible, is there any formula that can help me to take root from (for example) 1,2 without a calculator (by hand)? For example, there is a cos(x) formula that can be calculated on the paper: $$\cos x=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}$$ There is the Babylonian method...
  8. chwala

    Finding square root of number i.e. ##\sqrt{\dfrac{16}{64}}##

    The correct answer is; ##\sqrt{\dfrac{16}{64}}=\dfrac{4}{8}## . I do not seem to understand why some go ahead to simplify ##\dfrac{4}{8}## and getting ##\dfrac{1}{2}## which is clearly wrong. I do not know if any of you are experiencing this... I guess more emphasis on my part. Cheers! Your...
  9. EvilScientist

    B How Do You Calculate the 12th Root of x to the Fourth Power?

    I feel incredibly stupid for not getting this. I found this math problem in the beginning of my precalculus book: 12√x^4 That's 12th root of x to the fourth power. How do I find the root of x if the root is larger than the exponent?
  10. mopit_011

    B Derivative of Square Root of x at 0

    When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?
  11. e2m2a

    B Square Root of an Odd Powered Integer is Always Irrational?

    Is it always true that the square root of an odd powered integer will always be irrational?
  12. D

    Show that square root of 3 is an irrational number

    ##\sqrt{3}## is irrational. The negation of the statement is that ##\sqrt{3}## is rational. ##\sqrt{3}## is rational if there exist nonzero integers ##a## and ##b## such that ##\frac{a}{b}=\sqrt 3##. The fundamental theorem of arithmetic states that every integer is representable uniquely as a...
  13. F

    Repeated root in field of char 0

    Proof: We will first show ##\gcd(p(x), p'(x)) = 1##. Define ##d(x) = \gcd(p(x), p'(x))##. Then we can find ##q(x) \in F[x]## such that ##p(x) = d(x)q(x)##. But ##p(x)## is irreducible which means ##d(x)## is constant or ##q(x)## is constant. If ##q(x)## is constant, then ##\deg d(x) = \deg...
  14. e2m2a

    I Is the odd root of an even number always an irrational number?

    Is the odd root of an even number always an irrational number? For example the 7th root or the 11th root, etc. of an even number.
  15. S

    I Finding a polynomial that has solution (root) as the sum of roots

    AIUI, an algebraic is defined as a number that can be the solution (root) of some integer polynomial, and is any number that can be constructed via any binary arithmetic operation or unary root operation with arguments that are themselves algebraic numbers. I have been able to prove this for...
  16. A

    Proving this equation -- Limit of a sum of inverse square root terms

    Hi I was working on a physics problem and it was almost solved. Only the part that is mostly mathematical remains, and no matter how hard I tried, I could not solve it. I hope you can help me. This is the equation I came up with and I wanted to prove it: $$\lim_{n \rightarrow+ \infty} {...
  17. D

    Why does the numerator become 1 in the root test for solving series?

    in order to solve a series, the root test is applied and I have this limit ## \lim_{n \rightarrow +\infty} \sqrt[n] {\left| {\frac {i^{\frac { n} {3}}} { \frac {2n} {3} +1}} \right| } ## I don't understand why at the second step the numerator becomes 1, cannot recall why it becomes 1, that is...
  18. nomadreid

    I Shouldn't this definition of a metric include a square root?

    In https://mathworld.wolfram.com/InnerProduct.html, it states "Every inner product space is a metric space. The metric is given by g(v,w)= <v-w,v-w>." In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand, "As for every normed vector space, an inner product space is a metric...
  19. Brian1776

    PID - Root Locus (Sisotool) for Transfer Function with Zeros and Poles

    Hi everyone! I have a 8th order transfer function, you can see it in the first image: % Transfer function num = [2.091,0,203.3,0,-2151,0,-1.072e05]; den = [1,0,-830.4,0,-1.036e05,0,-5.767e05,0,2.412e07]; tf = tf(num, den) I need to use a PID, so I'm trying to use a compensator, adding poles...
  20. karush

    MHB Finding the Largest Root of a Polynomial Using Synthetic Division

    $\tiny{GRE.al.06}$ For the polynomial $x^3-3x^2-6x+8\quad -2$ is the smallest root. Find the largest root. $a.\, -1 \quad b.\, 1 \quad c.\, 2 \quad d.\, 3 \quad e.\, 4$ Since -2 is a root then use synthetic division $\begin{array}{r|rrrr} -2&1&-3&-6&8\\ & & -2& 10&-8\\ \hline &1& -5&...
  21. yucheng

    Computing a limit involving a square root: what is wrong?

    My attempt: \begin{align} \lim\limits_{n \to \infty} \sqrt{n^2 + n} - n &= n\sqrt{1+\frac{1}{n}} -n\\ &=n - n\\ &= 0\\ \end{align} I think the issue is at (1)-(2) For comparison, here is Rudin's solution
  22. Lilian Sa

    First order differential equation involving a square root

    Summary:: solution of first order derivatives we had in the class a first order derivative equation: ##\frac{dR(t)}{dt}=-\sqrt{\frac{2GM(R)}{R}}## in which R dependent of time. and I don't understand why the solution to this equation is...
  23. V

    B Determine a fractional square root without calculator

    I have to solve a certain numerical problem without using calculator and furthermore, there is a time limit for solving this problem. The answer I have got so far is ## \sqrt{\frac{100}{99}}## I know I can reduce the numerator to 10 but then I am stuck with square root of denominator which is...
  24. Mark44

    Fast reciprocal square root algorithm

    I ran into an interesting video on Youtube yesterday, about a fast way to compute the reciprocal of the square root of a number. I.e., to compute this function: ##f(x)= \frac 1 {\sqrt x}## The presenter is John Carmack. If you search for "Fast Inverse Square Root — A Quake III Algorithm" you'll...
  25. Y

    What Does 'Root' Mean in the Context of Installing Visual Studio?

    Hi I am trying to install all the necessary files for VS to run games, I just have a very simple question. This is the instruction. When the download completes, create a folder at the root of the same drive where you installed Visual Studio and name it SFML. Also, create another folder at the...
  26. Eclair_de_XII

    Proving that a fifth-degree polynomial has a root using just the IVT

    I consider three cases, based on the sign of ##a_0##. if ##a_0 == 0##: Set ##x=0##. \begin{align*} f(0)&=&a_0+a_1\cdot 0+a_2\cdot 0^2+a_3\cdot0^3+a_4\cdot0^4+0^5\\ &=&a_0+0\\ &=&0+0\\ &=&0 \end{align*} elif ##a_0<0##: Define ##M=\max\{|a_i|:1\leq a_i\leq 5\}## and set ##x=5(M+1)\neq 0##...
  27. anemone

    MHB Prove Root of Polynomial $P(x)=x^{13}+x^7-x-1$ Has 1 Positive Zero

    Prove that the polynomial $P(x)=x^{13}+x^7-x-1$ has only one positive zero.
  28. Strand9202

    Derivative of the square root of the function f(x squared)

    I started out by rewriting the function as (f(x^2))^(1/2). I then did chain rule to get 1/2(f(x^2))^(-1/2) *(f'(x^2). - I think I need to go further because it is an x^2 in the function, but not sure.
  29. M

    MHB Proving Boundedness of Seq. $(a_n)$ Given Bound Seq. $(\frac{a_{n+1}}{a_n})$

    Hey! :giggle: Show for each sequence $(a_n)\subset (0, \infty)$ for which the sequence $\left (\frac{a_{n+1}}{a_n}\right )$ is bounded, that $\sqrt[n]{a_n}$ is also bounded and that $$\lim \sup \sqrt[n]{a_n}\leq \lim \sup \frac{a_{n+1}}{a_n}$$ I have done teh following: The sequence $\left...
  30. Wrichik Basu

    Trouble installing MySQL on Ubuntu 20.04 — asks for root password

    I wanted to install MySQL on my laptop running Ubuntu 20.04, and was following this website for instructions. I executed the following commands: ~$ sudo apt update ~$ sudo apt install mysql-server ~$ sudo mysql_secure_installation After installation, I found that I could log into root user...
  31. C

    Troubleshooting ROOT Installation on MacOs

    ROOT is required as a pre-requisite for some software that I am trying to install. I'm on a MacOs system and I have tried to install using 'brew install root'. Do I need to do anything else? How can I check that root was successfully installed? When I tried to install said software, apparently...
  32. anemone

    MHB Can All Roots of a Quartic Polynomial Be Real?

    Let $a$ and $b$ be real numbers such that $a\ne 0$. Prove that not all the roots of $ax^4+bx^3+x^2+x+1=0$ can be real.
  33. sandmanvgc

    Square Root Practice: Multiplying by 1000NM/kJ

    Summary:: Why are you multiplying by 1000NM/kJ within square root? Practice problem for FE [Thread moved from the technical forums so no Homework Template is shown]
  34. LCSphysicist

    Find this sum involving a polynomial root

    if x_{I}, I = {1,2,...,2019} is a root of P(x) = ##x^{2019} +2019x - 1## Find the value of ##\sum_{1}^{2019}\frac{1}{1-\frac{1}{X_{I}}}## I am really confused: This polynomial jut have one root, and this root is x such that 0 < x < 1, so that each terms in the polynomial is negative. But the...
  35. Arman777

    I Writing Complex Roots of Negative Numbers

    Let us suppose I have a number ##x## such that ##x<0##. If I want to write the roots of the ##x^{1/n}##. How can we write the roots of this number. I thought we can write $$|x|^{1/n}e^{i\pi\theta}$$ for ##\theta = \frac{2l + 1}{n}## and ##l = 0,1,2## etc. Is this correct ? Similary If I...
  36. mcastillo356

    B Principal square root of a complex number, why is it unique?

    This is a quote from "Calculus", by Robert A. Adams. It's a translation from spanish: "Roots of square numbers If ##a## is a positive real number, there exist two different real numbers whose square is ##a##. They are ##\sqrt{a}\;## (the positive square root of ##a##) ##-\sqrt{a}\;## (the...
  37. M

    MHB Monotonically convergence to the root

    Hey! 😊 We have the following iteration from Newton's method \begin{align*}x_{k+1}&=x_k-\frac{f(x_k)}{f'(x_k)}=x_k-\frac{x_k^n-a}{nx_k^{n-1}}=\frac{x_k\cdot nx_k^{n-1}-\left (x_k^n-a\right )}{nx_k^{n-1}}=\frac{ nx_k^{n}-x_k^n+a}{nx_k^{n-1}}\\ & =\frac{ (n-1)x_k^{n}+a}{nx_k^{n-1}}\end{align*} I...
  38. AN630078

    Iterative root finding for the cube root of 17

    Firstly, the cube root of 17 is 2.571281591 which is 2.57 to 3.s.f. Initially, I thought about just approaching this problem using the Newton-Raphson Method when x0=2. In which case; x^3=17 x^3-17=0 Using the Newton-Raphson iterative formula xn+1=xr-f(xn)/f’(xn) f(x)=x^3-17 f’(x)=3x^2...
  39. J

    Using Equipartition theory to solve the root mean square of a angle.

    For the first question, i believe that mechanical energy is conserved hence we can derive the total energy i think. In regards to the second question, I'm assuming its at room temperature, so helium is monotonic therefore it has 3 degrees of freedom, therefore its internal energy is 3/2KT. I am...
  40. chwala

    Find the square root of a surd term

    find the square root of ## a+b+√(2ab +b^2)## let ##√[a+b+√(2ab +b^2)]= ±(√x +√y)## then, ##a+b+√(2ab +b^2)= x+y+ 2√(xy)## where ##a+b=x+y##.......1 ##(b+a)^2-a^2=4xy## .....2 from 2, ##a^2=(b+a)^2-4xy## ##a=√[x+y)^2-4xy]## ##a=√[x^2-2xy+y^2]## ##a=x-y## therefore...
  41. M

    MHB How to Solve an Equation with Square Roots?

    Please Help me solve it \[ \sqrt{x}+\sqrt{x+8}=8 \] thanks
  42. Rongeet Banerjee

    Root Mean Square Velocity of Gases

    This question came in NEET Exam 2018.Now my first query is that in the question,the mass of one Oxygen molecule is given wrong.Its exactly half it's true value.I don't think anybody has noticed this before because I couldn't find any change in the printed question on so many different books...
  43. tworitdash

    A Integral of a sinc squared function over a square root function

    I want to find the analytical solution to the integral given below. \int_{-\infty}^{\infty} \frac{ sinc^2(\frac{k_yb}{2})}{\sqrt{k^2 - k_x^2 - k_y^2}}dk_y In other words, \int_{-\infty}^{\infty} \frac{ \sin^2(\frac{k_yb}{2})}{(\frac{k_yb}{2})^2\sqrt{k^2 - k_x^2 - k_y^2}}dk_y Can this be...
  44. jk22

    B Is the sign of the square root dependent on the argument inside it?

    Could it be said that since ##a=A(f(x))\sqrt{f(x)}##, with ##A(x)\in\{1,-1\}## then ##a^2=f(x)##,, that ##a## is the square root of ##f(x)## ? In other words could the sign of the root depend on the argument inside it ? Else it would have to be chosen by human free will and to be blocked for...
  45. F

    I Physical meaning of the highest root / weight

    As some simple Lie groups and their algebras are essential for our current understanding of QM, I wondered if especially the highest positive (or likewise lowest negative) root can be explained physically. The roots are the weights of the adjoint representation. Are their physical meanings...
  46. anemone

    MHB Condition for A Quartic Equation to have a Real Root

    Show $20a^2+20b^2+5c^2\ge 64$ if $y=x^4+ax^3+bx^2+cx+4$ has a real root.
  47. Math Amateur

    MHB Complex Function Theory: Explaining Example 1.5, Section 1.2, Chapter III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I have yet another question regarding Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...
  48. Math Amateur

    MHB Complex Square Root Function: Qs from Bruce P. Palka's Ex. 1.5, Ch. III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need further help with other aspects of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...
  49. Math Amateur

    MHB Differentiating Complex Square Root Function: Bruce P. Palka, Ex. 1.5, Ch. III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with an aspect of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III, reads as...
  50. Math Amateur

    MHB Verify Gamelin's Remark: Complex Square and Square Root Functions

    I am reading Theodore W. Gamelin's book: "Complex Analysis" ... I am focused on Chapter 1: The Complex Plane and Elementary Functions ... I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ... The...
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