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.
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.
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?
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...
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...
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?
##\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...
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...
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} {...
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...
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
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...
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...
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...
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.
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]
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...
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...
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...
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...
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...
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...
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...
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 $$
From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...
Hi!
$$(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...
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...
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...
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...
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...
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...
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.
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...
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...
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 $\delta>0$...
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...
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...
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...
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...
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
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}...
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...
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...
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...
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?