What is Square root: Definition and 383 Discussions

In mathematics, a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 42 = (−4)2 = 16.
Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by



{\displaystyle {\sqrt {x}},}
where the symbol

{\displaystyle {\sqrt {~^{~}}}}
is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by



{\displaystyle {\sqrt {9}}=3,}
because 32 = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.
Every positive number x has two square roots:



{\displaystyle {\sqrt {x}},}
which is positive, and



{\displaystyle -{\sqrt {x}},}
which is negative. Together, these two roots are denoted as



{\displaystyle \pm {\sqrt {x}}}
(see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive x, the principal square root can also be written in exponent notation, as x1/2.Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.

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  1. Mohmmad Maaitah

    How to find range inside square root

    Hi, so I know how to find domain but how about range in this problem? I don't understand the way he did it? I always get answers wrong when it comes to range.
  2. 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?
  3. C

    Quadratic equation: Which way is correct? pic1 or pic2?

    I am a bit confused, so if anyone can explain to me which way is right I would be very thankful. I think that the way in pic 1 is right because of the properties written next to the procedure but the professor who posts videos on youtube solved it the way as written in pic 2 where he didn't...
  4. 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...
  5. BurpHa

    Find all possible solutions of x^3 + 2 = 0

    I actually know one way to solve, ##x ^ 3 + 2 = 0## ##x ^ 3 + \left (\sqrt[3] 2\right) ^ 3 = 0## ##\left(x + \sqrt[3] 2\right) \left(x ^ 2 - x\sqrt[3] 2 + \left(\sqrt[3] 2 \right)^2\right) =0## ##x + \sqrt[3] 2 = 0, x = -\sqrt[3] 2## ##x ^ 2 - x\sqrt[3] 2 + \left(\sqrt[3] 2\right)^2 = 0, x =...
  6. 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?
  7. 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?
  8. docnet

    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...
  9. A

    B It works but why? (Matching experimental data to a random equation)

    Hey guys, I've about a week left to submit my final paper for my trade degree in transportation. The paper is about an analysis of potential implementation of an electric car for direct deliveries in my area where I live. In part of it, I try to analyze how many possible trips a car like...
  10. 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} {...
  11. 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...
  12. 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
  13. 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...
  14. 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...
  15. 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...
  16. 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.
  17. 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]
  18. 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...
  19. 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...
  20. M

    MHB How to Solve an Equation with Square Roots?

    Please Help me solve it \[ \sqrt{x}+\sqrt{x+8}=8 \] thanks
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. I

    Epsilon delta proof of the square root function

    Let ##\varepsilon > 0## be arbitrary. Now define ##\delta = \text{min}\{\frac{a}{2}, \varepsilon \sqrt{a}\}##. Now since ##a>0##, we can deduce that ##\delta > 0##. Now assume the following $$ 0< |x-a| < \delta $$ From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...
  28. J

    I Element-wise square root of a vector notation?

    What is the notation to show element-wise square root of a vector or matrix?
  29. V

    MHB Square root n limit ( sum question )

    Hi! $$(x_{n})_{n\geq 2}\ \ x_{n}=\sqrt[n]{1+\sum_{k=2}^{n}(k-1)(k-1)!}$$ $$\lim_{n\rightarrow \infty }\frac{x_{n}}{n}=?$$ I know how to solve the limit but I don't know how to solve the sum $\sum_{k=2}^{n}(k-1)(k-1)!$ which should be $(n! - 1)$ The limit would become $\lim_{n\rightarrow \infty...
  30. Cheesycheese213

    How to find the square root of a square root?

    Homework Statement Simplify √(53 - 8√15) Homework Equations Numbers can be represented as √a - √b The Attempt at a Solution I had tried to make in an equation where the 2 expressions were equal, but after squaring both sides, I didn’t really know what to do. I had also tried to use something...
  31. srfriggen

    B Is the square root of 945 irrational?

    Is the square root of 945 irrational? I feel it is rational because my TI-84 Plus converts it into 275561/8964, however, I am unsure whether the calculator is estimating. Can someone please advise. It can be broken down into 3√105, and again, my calculator is able to convert √105 into a...
  32. DLeuPel

    How to get R= 2 ( square root ) h1h2

    Problem : A ball is let down a ramp on top of a table with initial velocity of 0 ms-1. When it reaches the end of the ramp, it is launched horizontally. Knowing that we don’t take air resistance or friction into account, and that the height of the ramp is h1, and that of the table is h2...
  33. DLeuPel

    I How to get R= 2 ( square root ) h1h2

    A ball is let down a ramp on top of a table with initial velocity of 0 ms-1. When it reaches the end of the ramp, it is launched horizontally. Knowing that we don’t take air resistance or friction into account, and that the height of the ramp is h1, and that of the table is h2 relative to ground...
  34. Gionata

    I Recursive square root inside square root problem

    I have been debating this issue for days: I can't find a recursive function of this equation: ##\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}## Starting value 2 always added with pi has been trying to find a solution this for days now, is what I have achieved so far: This...
  35. pairofstrings

    B Why to write numbers in square roots and not in decimals?

    Hi. I have coefficient of x2 as in an expression that looks like this * calculator shows little yellow triangle because 'x' is not defined. If I can write the coefficient of x2 as - 0.091372213746554 then why did the author write coefficient of x2 like this shown below? Thanks.
  36. R

    I "Undo" Second Derivative With Square Root?

    In my classical mechanics course, the professor did a bit of algebraic wizardry in a derivation for one of Kepler's Laws where a second derivative was simplified to a first derivative by taking the square root of both sides of the relation. It basically went something like this: \frac{d^2...
  37. Poetria

    Approximating square root of 2 (Taylor remainder)

    Homework Statement [/B] Use the Taylor remainder theorem to give an expression of ##\sqrt 2 - P_3(1)## P_3(x) - the degree 3 Taylor polynomial ##\sqrt {1+x}## in terms of c, where c is some number between 0 and 1 Find the maximum over the interval [0, 1] of the absolute value of the...
  38. C

    MHB Find Limit of $\sqrt{x}$ as $x\to c$, $c\ge 0$

    Dear Everybody, I am having trouble to determine the value of delta when c is strictly greater than 0. Here is the work: The Problem: Find the Limit or prove that the limit DNE. $\lim_{{x}\to{c}}\sqrt{x} for c\ge0$ Proof: Case I: if c>0. Let $\varepsilon>0$ Then there exists $\delta>0$...
  39. Allan McPherson

    Approximating Damped Oscillator Time Period and Frequency with Large n

    Homework Statement An oscillator when undamped has a time period T0, while its time period when damped. Suppose after n oscillations the amplitude of the damped oscillator drops to 1/e of its original value (value at t = 0). (a) Assuming that n is a large number, show that...
  40. Adgorn

    B Square root of a negative number in a complex field

    Mod note: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { } (This question is probably asked a lot but I could not find it so I'll just ask it myself.) Does the square root of negative numbers exist in the complex field? In other words is...
  41. Mr Davis 97

    I Proving that square root of 2 exists

    I am reading Abbot's "Understanding Analysis," and in this text he assumes that the real numbers are complete, that is, he assumes the least upper bound property, and begins to prove everything from there. Later in the book he proves that the square root of 2 does in fact exist in...
  42. N

    I Square root of a complex number

    if a is a complex number then sqrt(a^2)=? Is it is similar to Real Number? Help me please
  43. L

    B Representation of complex of square root of negative i with unitary power.

    Can ##sqrt(-i)## be expressed as a complex number z = x + iy with unitary power?
  44. I

    Arc Length of Parabola & Square Root Function

    Homework Statement Consider the curves: y = x^2 from 1/2 to 2 and y = \sqrt{x} from 1/4 to 4. a. Explain why the lengths should be equal. b. Set up integrals (with respect to x) that give the arc lengths of the curve segments. Use a substitution to show that one integral can be...
  45. K

    B Square root differential problem

    Hi, I working on their text this equation did not make sense to me. From equation 1 it differentiate second term , I wonder how he got second term of equation 2. What I think is, what I wrote at the bottom
  46. Mr Davis 97

    I Proof that the square root of 2 is irrational

    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}...
  47. Adgorn

    Proving the square root of a positive operator is unique

    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...
  48. H

    B Understanding r^2 and the Role of Square Root in Data Analysis

    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...
  49. binbagsss

    Integration question involving square root

    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...
  50. M

    MHB Square Root vs Cube Root

    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?