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.
Homework Statement
Finding derivative of (sin(sqt3x+5))
Homework Equations
None given. Chain Rule
The Attempt at a Solution
The answer is: (cos(sqrt3x+5)) * 1/2(sqrt3x+5) * 3
but I don't know how to get to the 3.
I turned sin into cos and multiplied by the inside derivative giving the...
Homework Statement
Let e be the number close to sqrt(a) by Newtons Method (That is picking a number, diving a by it, and taking their average, divide a by average, get a number, find their average, so on). Using |e<sqrt(a)+e|
prove that if |a/e-e|<1/10
then |sqrt(a)-e|<1/10
Note that e is...
Hi,
I have a quick question about making quantum mechanics relativistic by simply replacing the hamiltonian by a relativistic hamiltonian. If we write the hamiltonian operator as:
H = \sqrt{P2c2 + m2c4},
schrodinger's equation in position basis becomes:
i\hbar\dot{\psi} =...
If I have a matrix valued function A(x) of some scalar x, how do I compute the derivative of the square root of A with respect to x? It seems like it should be simple, but I can't find it anywhere on the internet. Thanks!
Is the answer to sqrt -81y^3 : y sqrt-81y? or is there no real solutions?
Also for this radical equation:
sqrt 2n-5 - sqrt 3n+4=2
I worked it out and can't seem to get an answer. Is there no real solutions?
At first I thought that there is no square matrix whose square is the 0 matrix. But I found a counterexample to this. My counterexample is:
\left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right)
However it appears that my counterexample has a 0 row. I'm curious, must a square root of...
Homework Statement
Ok, working on a inverse function question, and I got stuck with something.
Can someone explain the steps that makes this possible here. Something I am missing :(
f(x) = √(x^3 + x^2 + x + 1)
How is the inverse of the above function this...
3x^2 + 2x + 1 / 2√(x^3 + x^2 +...
Homework Statement
I know how to prove that square root of 2 is irrational using the well ordering principle but what I'm wondering is, how can we use the well ordering principle to prove this when the square root of two isn't even a subset of the natural numbers? Doesn't the well ordering...
If you could see the image attached, I think it looks better than me typing it here. Didn't know how to embed the image.
I would just like to know how it becomes 2v to √2v.
EDIT: Ignore. Figured it out. Don't know why I was even baffled. :/
Provided is a function f(x)=\sum_{j=1}^n ||x-x_j||, for x being a two dimensional vector, where ||.|| denotes the Euclidean distance in 2D space. How could one obtain a derivative of such a function?
Hi. I just wondered why we use a 1/\sqrt{V} in the Fourier expansion of the vector potential. A regular 3 dimensional Fourier expansion is just
f(\vec r) = \sum_{\vec k} c_\vec{k} e^{i \vec k \cdot \vec r}
but as the solution to the equation
(\frac{\partial ^2}{\partial t^2} -...
I have to find the limit as x→∞ of √(x2+x)-xI can't rearrange this into a form where I can put infinity into the expression and get a meaningful answer. I've tried taking out square roots to get √x( √(x+1)-√x ) but if I put infinity into this I just get ∞(∞-∞) which is meaningless.
Now I know...
Homework Statement
∫(e^x)/(√4-e^(2x))
Homework Equations
arcsin of x
The Attempt at a Solution
I know how the problem should be solved and have an idea of what the final answer will be. My only question is, how would I take out the four from the square root, in order to make it...
Homework Statement
Find dy/dx when y=\sqrt{ln x}
Homework Equations
d/dx of ln x is equal to 1/x times d/dx of x.
The Attempt at a Solution
I tried to raise the ln x to the 1/2 power instead of keeping it under a square root sign, but I had no luck. I'm struggling with Calculus. I...
Its been a while since I have taken any kind of math class, I am a bit rusty in general algebra. Can someone explain how I would multiply an equation like this
(2x-1)sqrtof x-3x
is it just like normal distribution? Would I just put the answer underneath the square root?
sqrt2x^2-6x^2-x+3x?
Homework Statement
I have the function H^2 + 12756H and I want to find the domain and range of it's square root algebraically. Homework EquationsThe Attempt at a Solution
I understand y= √(H^2 + 12756H) is undefined if H^2 + 12756H < 0, however I don't get how to find its domain and range...
Hi. I'm currently reading about (negative frequency) solutions to the Dirac equations which can be written on the form
\Psi = ( \sqrt{p \cdot \sigma} \chi, \sqrt{p \cdot \bar{\sigma}} \chi)^T e^{-i p \cdot x}For any two component spinor Chi. But the dot product with the four vector p and the...
Finding Limit As "X" Approaches Infinite Of Square Root Function
Homework Statement
Homework Equations
None that I am aware of.
The Attempt at a Solution
What I tried to do to solve this problem was first, multiplying the function by its conjugate, and then simplifying the...
Homework Statement
The image has the question I don't quite understand!
Homework Equations
The Attempt at a Solution
I understand how to get √(√2 - x) but I don't get how they end up with: 4√2 - x
?
Hoping someone can push me in the right direction with this one. Plume snookered.
It's to simplify:
2√3(3+√3)
Guessing first calculate (a^2 - b^2*c) in the square, though the 2 is throwing this an I'm not sure how the answer is 6√3 + 6, an not 18√3 ?
Homework Statement
Prove Square Root of 15 is Irrational
The Attempt at a Solution
Here's what I have. I believe it's valid, but I want confirmation.
As usual, for contradiction, assume 15.5=p/q, where p,q are coprime integers and q is non-zero.
Thus, 15q2 = 5*3*q2 = p2...
IDK if this should be in the precalc section, but I was wondering how to reduce
\sqrt{(3+\sqrt{5})} / \sqrt{(3-\sqrt{5})} to (\sqrt{5} + 1) /(\sqrt{5} - 1)
In what branches of mathematics is this proven.. I have never seen a proof, so I wonder if anyone can give me the basics of what is done to proove it or got a link to a proof..
Edit: By square root I mean the positive square root.
Suppose I have to solve for y:
x\leq 1
(x - 1)^{2} = y
So I know that (x - 1) will always be 0 or a negative, therefore I must take the negative square root of (x - 1):
-\sqrt{(x - 1)^{2}} = -\sqrt{y}
Am I to understand that this is the same as:
-1 \cdot \sqrt{(x - 1)^{2}} = -1 \cdot...
Let $a,b \in \mathbb{Z}$, and if $a+b\sqrt{2}$ has a square root in $\mathbb{Q}(\sqrt{2})$, then the square root is actually in $\mathbb{Z}[\sqrt{2}]$.
Only one approach comes to my mind. Let $r_1, r_2 \in \mathbb{Q}$ such that $a+b\sqrt{2}=(r_1+r_2\sqrt{2})^2$. This gives $a=r_1^2+2r_2^2...
I was playing trying to work through a proof in Apostol's Calculus and can't quite understand a step noted. This is from chapter 3, theorem 1.35. Every nonnegative real number has a unique nonnegative square root. The part where you are establishing the set S as nonempty so you can use LUB it...
Years back i learned the digit-by-digit calcualtion of square root like this : http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Digit-by-digit_calculation
But i don't know why this works. Wikipedia gives kind of an explanation but i don't understand it. How does this method work?
Homework Statement
Integrate:
sqrt(1/4 + t^2 + t^4)The Attempt at a Solution
I'm really not sure on how to go about integrating this, it's actually integrate from -1 to 1, the solutions manual has a method I'm not familiar with. I thought of factorising it first, although doing that hasn't...
line integral -- confusion on squares and square root terms
Homework Statement
Do you see where they have sqrt(16 sin^2t etc = 5? How do they get that, the answer should be 7, the square root of 16 is 4, sin^2 + cos^2 is 1 and the square root of 9 is 3, 3 + 4 = 7. It's like they're...
By chance I stumbled on this "almost" equality:
\frac{1}{5}(1/2+2/3+3/4+4/5+5/6+6/7+7/8+8/9+9/10) ≈ √2 - 7.2×10^{-6}
I'm just wondering, are these funny coincidences simply, well, coincidences :biggrin: or is there some kind of explanation?
I've see a ton of other funny stuff like...
I am trying to prove sqrt(3) is irrational. I figured I would do it the same way that sqrt(2) is irrational is proved:
Assume sqrt(2)=p/q
You square both sides.
and you get p^2 is even, therefore p is even.
Also q^2 is shown to be even along with q.
This leads to a contradiction.
However...
Homework Statement
Define f(x)=\sqrt{}(x + (\sqrt{}(x + \sqrt{}x)
Determine where f is differentiable and compute the derivative
Homework Equations
f'(xo)= lim as x approaches xo (f(x) - f(xo))/(x - xo)
The Attempt at a Solution
By the definition, f(x) = \sqrt{}x does not have a...
Homework Statement
The problem and solution are included.
Homework Equations
Double integration.
The Attempt at a Solution
Firstly, I'd like to mention that the additional ρ under the square root is there accidentally and that it should be outside of the square root such that it...
I'm having a little bit of trouble figuring out how exactly to do this.
Prove that \sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}}} = \frac{1\pm\sqrt{4n+1}}{2}.
How exactly does one go about doing this? I mean, I understand it goes on infinitely, but doesn't that create an infinitely...
Homework Statement
Let a be a positive real number. Prove that if a is irrational, then √a is
irrational. Is the converse true?
Homework Equations
So, an irrational number is one in which m=q/p does not exist. I understand that part, but then trying to show that the square root of an...
Homework Statement
Solve \sqrt{-e^{(i2\pi)/3}}
Homework Equations
The Attempt at a Solution
I seem to be missing something simple, as I take:
\sqrt{-1} = i
then,
e^{(1/2)*(i2\pi)/3}
which comes out as: ie^{i\pi/3}
however, the solution is:
-ie^{i\pi/3}, and I can't seem to see where...
Homework Statement
z = (n + i)^{2}
n is a positive real number, and arg(z) = \frac{\pi}{3}
Find the value of n.
The attempt at a solution
I am reviewing old problem sets from years past, and came across this problem that appears pretty simple. I have my old answer as n=\sqrt{3}...
why there is no answer when negative value is being square root?
e.g: square root of -9
when i try to find answer from calculator, ''math error '' appears..
so is there an explanation for this question??
this question may looks so weird..but i m juz asking out of curiousity..
Can a kind person explain to me why I appear to have two conflicting solutions to:
\int^{\frac{1}{\sqrt{2}}}_0dx\sqrt{1-x^2}
Solution 1 : Standard trigonometric substitution: x=\sin\theta
Integral becomes...
Please see attached. This is a general question for a more complex thermo problem, but it fits this form. Are there any tricks to getting expressions out of square roots in solving DEs?
y=y(x)
x=x(t)
I want to get y by itself so I can integrate with respect to x, but it is trapped in a...
Homework Statement
What I am confused about is:
x - \sqrt{2}=0
x=\sqrt{2}
is it equal to
x=\pm\sqrt{2}
?
Homework Equations
The Attempt at a Solution
The way I know it, those two are not equal. But my teacher insists on saying that whenever there is a square root, we should always...
Homework Statement
Let f be a quadratic polynomial function of z with two different roots z_1 and z_2. Given that a branch z of the square root of f exists in a domain D, demonstrate that neither z_1 nor z_2 can belong to D. If f had a double root, would the analogous statement be true?Homework...
Homework Statement
Let A be the matrix: -5 -3
18 10
Find an invertible matrix X so that XAX-1 is diagonal. Use this to find a square root of the matrix A.
Homework Equations
DetA - xI
(A-\lambdaI)v = 0
The Attempt at a Solution
So, I found DetA- xI, which...
In Ramakrishna's paper, http://arxiv.org/ftp/arxiv/papers/1111/1111.1922.pdf , he derived equation (9), page 5.
It is a difference of two square-roots using an approximation method. Can anyone help in how this is done?
Thanks