What is Roots: Definition and 976 Discussions

The Roots are an American hip hop band, formed in 1987 by Tariq "Black Thought" Trotter and Ahmir "Questlove" Thompson in Philadelphia, Pennsylvania, United States. The Roots serve as the house band on NBC's The Tonight Show Starring Jimmy Fallon, having served in the same role on Late Night with Jimmy Fallon from 2009 to 2014.
The Roots are known for a jazzy and eclectic approach to hip-hop featuring live musical instruments and the group's work has consistently been met with critical acclaim. ThoughtCo ranked the band #7 on its list of the 25 Best Hip-Hop Groups of All-Time, calling them "Hip-hop's first legitimate band."In addition to the band's music, several members of the Roots are involved in side projects, including record production, acting, and regularly serving as guests on other musicians’ albums and live shows.

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  1. A

    Complex numbers and negative roots

    I was wondering if scientists or mathematicians have any use for complex numbers involving negative roots of I as in i=(-1)^(1/2). but my question is more what would be (-1)^(-1/2)?
  2. ognik

    MHB Finding Complex Roots: Poles of $ \frac{1}{{z^4}+4} $, {z: |z-1| LE 2}

    I think I'm a bit rusty here, started with finding poles for $ \frac{1}{{z^4}+4} $, {z: |z-1| LE 2} 1) Out of interest, is there a complex equivalent of the rational roots test? The above function is obvious, but for a poly that has both real and complex roots? 2) I am using the exponential...
  3. Albert1

    MHB Roots of Equations & Sum of Inverses: $a=1,2,3,\dots,2011$

    $a=1,2,3,4,5,------2011$, the roots of the equations $x^2-2x-a^2-a=0,$ are : $(\alpha_1,\beta_1),(\alpha_2,\beta_2),----------,(\alpha_{2011},\beta_{2011})$ respectively please find : $\sum_{n=1}^{2011}(\dfrac{1}{\alpha_n}+\dfrac {1}{\beta_n})$
  4. A

    Alternative path to taking roots of both sides of equation

    The full question is: "How can we take square root of both sides of an inequality or equation just by multiplying each side by numbers with negative rational exponents". I will include several examples to explain how I think about it. 1)a=b, a^(-0.5)*a=b*a^(-0.5) (but a^(-0.5)=b^(-0.5)) then...
  5. ognik

    MHB Find Roots of sin z: Solutions & Explanations

    Looking for someone to check my working & answers please. Problem is 'find all the zeros of sin z' A) sin z = sin(x+iy) = sin(x)cosh(y) + i cos(x)sinh(y) Roots are when sin(x)cosh(y) = 0 = cos(x)sinh(y) $If \: sinh(y)=0, then \: cosh(y)=1 \: (cosh^2 - sinh^2=1) $ $ \therefore sin(x) = 0...
  6. M

    Roots of Negative Numbers (Complex Analysis)

    Homework Statement Express (-1)1/10 in exponential form (My first time posting - I hope I got the syntax right!) Homework Equations The Attempt at a Solution [/B] I got the solution, it's ejπ/10, but I'm not sure why. Here's my work: (-1)1/10 = (cos(π) + jsin(π))1/10 = cos(pi/10) +...
  7. Destroxia

    2nd Order Homogeneous, Real Roots, Initial Value

    Homework Statement Solve the initial value problem Homework Equations Quadratic Formula The Attempt at a Solution My problem is that I don't understand how to solve the constants now, I understand, 2 equations, 2 unknowns, but when I plug the y(0) = 0 into the YsubH equation...
  8. S

    Irrational Roots Theorems for Polynomial Functions

    Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found...
  9. anemone

    MHB Summing a Series of Cubic Roots

    Sum the series below: \sum_{n=1}^{999}\dfrac{1}{a_n} where $a_n=\sqrt[3]{n^2-2n+1}+\sqrt[3]{n^2+2n+1}+\sqrt[3]{n^2-1}$.
  10. M

    When do roots of a polynomial form a group?

    I've been studying for my final exam, and came across this homework problem (that has already been solved, and graded.): "Show that the Galois group of ##f(x)=x^3-1## over ℚ, is cyclic of order 2." I had a question related to this problem, but not about this problem exactly. What follows is...
  11. Jaco Viljoen

    Without solving the equation show it has 2 rational roots

    Homework Statement Without solving the equation 3x^2-8x-3=0 show it has 2 different rational roots.[/B]Homework Equations ax^2+bx+c=0 The Attempt at a Solution I would appreciate if someone would check my work, and advise if I have done the right or wrong thing? Thank you, Jaco [/B]...
  12. M

    MHB Field extensions and roots of polynomials

    Let F be a field extension of Q (the rationals) with [F:Q] = 24. Prove that the polynomial x^5+2x^4-16x^3+6x-10 has no roots in F. Proof: Let a be a root of x^5+2x^4-16x^3+6x-10. Since the polynomial has degree 5 by theorem we know that [Q(a):Q]=5. If a \in F and [F:Q]=24 then by theorem we...
  13. A

    MATLAB Roots of Polynomials by loop in matlab

    Dear Friends! I need to find roots of polynomials with variable coefficients, The command I used is w=0:50 A=w^2 B=w^3+2 C=w+2*w^2 D=w E=w./2 ss=[A B C D E] xi=roots(ss) by this I find all the roots of equation, I want to find velocities by setting v1=w/xi(1) v2=w/xi(2) v3=w/xi(3) v4=w/xi(4)...
  14. C

    MHB How to distribute square roots without making common mistakes?

    I'm homeschooled, but it's gotten to the point that my Mom doesn't know how to do what she's teaching me anymore. So now I'm teaching myself with just a textbook and no one to explain it to me. I'm stuck on an issue probably simple, but I still need help. I believe I messed up on the last lines...
  15. mooncrater

    A cubic equation and its roots

    Homework Statement The question says that : Find the value of ##a## so that the equation $$x^3-6x^2+11x+a-6=0$$ has exactly three integer solitions. Homework Equations IF ##p##,##q##,##r## are the roots of this equation then: ##p+q+r=6## ##pq+pr+rq=11## ##pqr=6-a## The Attempt at a Solution I...
  16. Jaco Viljoen

    For what values of k will the equation have no real roots?

    Homework Statement 2x^2-3x+kx=-1/2 1. k<1 or k>1 2. 1<=k<=5 3. k<=1 or k>=5 1<k<5 Homework Equations b^2-4ac a=2 b=3 c=k The Attempt at a Solution (3)^2-4(2)(k) =9-8k<0 =9/8<k =1&1/8<k I get the answer above but don't know how it relates? Any insight would be appreciated. Thank you, Jaco
  17. M

    MHB Square Root Solutions for Complex Numbers

    Helppp for part (ii). I got 3$e^{\frac{1}{6}\theta i}$
  18. N

    Repeated complex conjugate roots for Cauchy-Euler

    Looking for the general equation for repeated complex conjugate roots in a 4th order Cauchy Euler equation. This is incorrect, but I think it is close: X^alpha [C1 cos(beta lnx) + C2 sin(beta lnx)^2] I think that last term is a little off. Maybe C2 sin [beta (lnx)] lnx ?
  19. ichabodgrant

    Decomposition using roots of unity

    Homework Statement Decompose x5 - 1 into the product of 3 polynomials with real coefficients, using roots of unity. Homework Equations As far as I know, for xn = 1 for all n ∈ ℤ, there exist n distinct roots. The Attempt at a Solution [/B] So, let ω = e2πi/5. I can therefore find all the 5th...
  20. N

    Factoring equation with real coefficients

    Homework Statement Find the roots of z^4+4=0 and use that to factor the expression into quadratic factors with real coefficients. Homework Equations DeMoivre's formula. The Attempt at a Solution I have been able to identify they are \pm 1 \pm i but i have no idea how to factor the...
  21. A

    MHB Write equation given roots

    Hi everyone. I was given a problem in which the roots of a quadratic function were given. Using those roots, I had to write the quadratic function, with integer coeffecients only. The roots were: (-1+ (sqrt -2))/5 and its conjugate. The equation I have so far is: f(x) = 5(x^2) + 2x + (3/5)...
  22. alexmahone

    MHB Primitive Roots Modulo $p$: The $(p-1)/2$ Rule

    Is it true that $g$ is a primitive root modulo $p$ if and only if $g^{(p-1)/2} \equiv -1 \pmod p$?
  23. S

    MHB Square Roots Calculation Tricks

    hi all... i have problem about square roots for fast calculation, like below sample : is there fast calculation method not commonly/usually ways. it's possible? please, see my picture? thanks in advance.. susanto
  24. A

    Roots of series of exponential raised to power of x?

    How to solve: a1e-k1x+a2e-k2x+...+ane-knx =0 for x? For example in simple case of n=1,2. a1e-k1x+a2e-k2x=0 the solution will be x=In (a1/a2) / [ k1-k2]. But for terms >2 what will be the solution?
  25. anemone

    MHB Finding $b(a+c)$ Given Roots of a Cubic Equation

    let $a>b>c$ be roots of $\sqrt{2015}x^3-4031x^2+2=0$, find $ b(a+c)$.
  26. Shackleford

    Find the square roots of a = root3 + root3*i

    I don't recall ever doing this but maybe I have. z2 = a = p [cos Ψ + i sin Ψ] = √3 + i*√3 p = √6 Ψ = π/4 Using the formula in the notes, z = 61/4 * exp[i*(π/4 + 2π*k)/2], k = 0, 1.
  27. anemone

    MHB Can the Roots of a Cubic Equation be Bounded?

    Let $p,\,q,\,r$ be real numbers such that the roots of the cubic equation $x^3+px^2+qx+r=0$ are all real. Prove that these roots are bounded above by $\dfrac{2\sqrt{p^2-3q}-p}{3}$.
  28. R

    Integral with roots on bottom and top

    It's the integral of sqrt(x)/(cubed root(x) + 1) I tried regular u substitution but that didn't let me get rid of all the x's. I also just tried long division but that gave me an answer that didn't match with the actual answer to the problem. The actual answer is 6[1/7 x^(7/6) - 1/5 x^(5/6) +...
  29. kaliprasad

    MHB Find Real Roots of $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$

    find the real roots of the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1 $
  30. P

    Why is the principal square root of a complex number not well-defined?

    Within the context of real numbers, the square root function is well-defined; that is, the function ##f## defined by: ##f(x) = \sqrt{x}## Refers to the principal root of any real number x. Is it true that this is not the case when dealing with complex numbers? Does ##\sqrt{z}##, where ##z ∈ ℂ##...
  31. C

    Constructability of Roots (esp w/Hilbert Tools)

    ello all, This is largely a repost of something I posted on r/math, but didn't seem to find any luck there with two of my questions, so I'm asking it again here with the hopes that someone here can answer my questions. Thanks! I'm studying for a final in geometry, and I know I'm going to get a...
  32. ELB27

    A strange inconsistency with square roots

    Hi, I have a question that came into my mind while solving some problems. If I have a constant times an expression in a square root like ##4\sqrt{16}## I can square the constant and push it into the square root: ##4\sqrt{16}=\sqrt{4^2 16} = 16##. But what if the constant outside of the square...
  33. N

    Find Roots of Function & Return Column Vector

    i need to create a function that returns the pure zeros on the left semiplane and 0 if there is one and only one zero in the origin of the referential. the return has to be in a column vector like [root1;root2;...;rootn;0] or [root1;root2;...;rootn] if there is no root in the origin of the...
  34. B

    High precision square roots

    Hello, i'm having trouble evaluating my gamma factor for my special relativity homework, because I need to compute 1 minus a very small number (8.57*10^-13). My calculator treats this value as simply 1, as does Mathematica. Although I don't know much about it, and maybe there's a way to force...
  35. anemone

    MHB Find the sum of the real roots

    Find the sum of the real roots for $2x^8-9x^7+20x^6-33x^5+46x^4-66x^3+80x^2-72x+32=0$.
  36. A

    MHB CTS and show the roots in this form

    I have to show the roots of x^{2}-8x-29=0 are c\pmd\sqrt{5} I used completing the square method. Once I used CTS I got the answer (x-4)^2-45=0 So I am not sure what is the next step to put it in the form of c\pmd\sqrt{5}
  37. PcumP_Ravenclaw

    Roots of unity, Roots of complex equations of the form z^n = 1

    Dear all, please see the page above, (Alan F, Beardon, Abstract Algebra and Geometry). On the page, Theorem 3.5.2 says that the set of Complex numbers from ## z^n = 1 ##, where ## |z| = 1 ## forms a group w.r.t multiplication. I want to know if... The inverse of all elements...
  38. B

    Roots of the normal distribution

    Homework Statement $$f:\mathbb{R} \rightarrow \mathbb{R},$$ $$ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2 \sigma ^{2}}}$$ What are the roots of this equation? Homework EquationsThe Attempt at a Solution The roots of an equation are the values of x such that f(x) = 0. This...
  39. A

    RPM Effects on CFM & Pressure of a Rotary (Roots) Blower

    Hi, I would like to know how the RPM of a root blower will affect the CFM and pressure created by the blower. The confusion that I have is this: Generally when RPM is decreased, both the CFM and pressure decreases. But according to Bernoulli's principle, should not one decrease and the other...
  40. U

    Rudin PMA Theorem 1.21 Existence of nth roots of positive reals

    Homework Statement For every real x>0 and every n>0 there is one and only one positive real y s.t. yn=x Homework Equations 0<y1<y2 ⇒ y1n<y2n E is the set consisting of all positive real numbers t s.t. tn<x t=[x/(x+1)]⇒ 0≤t<1. Therefore tn≤t<x. Thus t∈E and E is non-empty. t>1+x ⇒ tn≥t>x, s.t...
  41. E

    Imaginary Roots and Vieta: 3a < 2+4c

    Homework Statement If both roots of the equation ax^2 + x + c - a = 0 are imaginary and c > -1, then: Ans: 3a < 2+4c Homework Equations Discriminant < 0 for img roots Vieta The Attempt at a Solution 1-4(a)(c-a)<0 4ac > 4a^2 + 1 Minimum value of 4a^2 + 1 is 1 so 4ac>1 I can't think of...
  42. _N3WTON_

    Second Order ODE, Complex Roots, Change of Variables

    Homework Statement Solve: \frac{d^{2}y}{dx^{2}} + \omega^{2}y = 0 Show that the general solution can be written in the form: y(x) = A\sin(\omega x + \alpha) Where A and alpha are arbitrary constants Homework EquationsThe Attempt at a Solution I know that I will need to change variables for...
  43. R

    MHB Show that Q adjoin square roots of 2, 3 is a vector space of dimension 4 over Q

    Let \mathbb{Q}(\sqrt{2},\sqrt{3}) be the field generated by elements of the form a+b\sqrt{2}+c\sqrt{3}, where a,b,c\in\mathbb{Q}. Prove that \mathbb{Q}(\sqrt{2},\sqrt{3}) is a vector space of dimension 4 over \mathbb{Q}. Find a basis for \mathbb{Q}(\sqrt{2},\sqrt{3}). I suspect the basis is...
  44. S

    Finding roots for a root locus

    Hi, In the attached image the roots are shown for the characteristic equation. I don't know how the roots were found. Anyone able to help? Thanks Splint
  45. anemone

    MHB Show all real roots are negative

    Show that all real roots of the polynomial $f(x)=x^5-10x+38$ are negative. Note: I know this is a fairly easy challenge, but it's good to see how different approaches can be generated from different people so that we can learn from one another. :o (Yes)
  46. T

    MHB Finding limit of a funciton with square roots.

    I have to find this: $$\lim_{{x}\to{3}}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3}$$ So I do this: $$\lim_{{x}\to{3}}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3} * \frac{\sqrt{6x + 14} + \sqrt{x+1}}{\sqrt{6x + 14} + \sqrt{x+1}}$$ The top part is easy since $$(\sqrt{a} - \sqrt{b})(\sqrt{a} +...
  47. M

    Inverse laplace transform (polynomial division? Complex roots?)

    Homework Statement Decide the inverse laplace transform of the problem below: F(s)= \frac{4s-5}{s^2-4s+8} You're allowed to use s shifting. Homework Equations The Attempt at a Solution By looking at the denominator, I see that it might be factorized easily, so I try that...
  48. T

    Exploring Complex Roots of Underdamped Systems: Why the Sine Term?

    I'm trying to do some refreshing of differential equations featuring damped systems. Specifically, I have a question regarding the differential equation solution to an under damped system involving complex roots. Referring to the attached pdf, an under damped system will yield a complex...
  49. P

    Mastering Simplification of Square Roots with Multiple Terms

    How do I solve, 1: (√3*√3*√3)/(√3+√3+√3)? How do I simplify it? I'm confused on how to shorten √x+√x+√x, I just don't get it. Also if: √700 = 26.46, then how is √70000 = 264.6? Shouldn't it be 2646?
  50. M

    Internal semidirect product and roots of unity

    Homework Statement Let ##G=G_{12}##, ##H_1=G_3##, ##H_2=G_2##. Decide if there are groups ##K_1##, ##K_2## such that ##G## can be expressed as the internal semidirect product of ##H_i## and ##K_i##.The Attempt at a Solution Suppose I can express ##G_{12}## as an internal semidirect product...
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