# 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

x

,

{\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

9

=
3
,

{\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:

x

,

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

x

,

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

±

x

{\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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

39. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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?