Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n, and pronounced as "b raised to the power of n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:
b
n
=
b
×
⋯
×
b
⏟
n
times
.
{\displaystyle b^{n}=\underbrace {b\times \dots \times b} _{n\,{\textrm {times}}}.}
The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the nth power", "b raised to the power of n", "the nth power of b", "b to the nth power", or most briefly as "b to the nth".
One has b1 = b, and, for any positive integers m and n, one has bn ⋅ bm = bn+m. To extend this property to non-positive integer exponents, b0 is defined to be 1, and b−n (with n a positive integer and b not zero) is defined as 1/bn. In particular, b−1 is equal to 1/b, the reciprocal of b.
The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.
Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.
Hi,
I have problems with task b, more precisely with the calculation of the limit value:
By the way, I got the following for task a ##f^{(n)}(x)=(-1)^{n+1} \frac{(n-1)}{x^n}##
Unfortunately, I have no idea how to calculate the limit value for the remainder element, since ##n## appears in...
I'm reading a book called Asymptotic Methods and Perturbation Theory, and I came across a derivation that I just couldn't follow. Maybe its simple and I am missing something. Equation 3.3.3b below. y(x) takes the form A(x)*(x-x0)^α and A(x) is expanded in a taylor series.
I am currently reading this paper where on page 8, the authors say that:
This correlates with Figure 8 on page 12.
Does it mean that there is a real correlation between eigenvalues and Lyapunov exponents?
Problem statement : If ##\boldsymbol{\displaystyle x^{\displaystyle 2x^{\displaystyle 6}}=3}##, evaluate ##\boldsymbol{\left(x^{\displaystyle x^{\displaystyle x^{\displaystyle 6}}}\right)^{\displaystyle\sqrt 3} = ?}##.
Attempt : I copy and paste my attempt using Autodesk...
This equation takes a present value (PV) to find mortgage payments, PMT:
Alternatively, switching V for PV and T for PMT:
V/T = r(1-r^n)/(1-r)
What is an algebraic method to solve for "r"?
Can it not be solved for? I realize I can just find out "r" by trial by error in Excel using the PMT...
Hello.
I wanted to construct a simple and clear explanation for the relation between the beta function and the critical exponents (divergence of correlation length) close to a critical point.
I wanted to check if this reasoning is valid. This is my own rewording of complicated arguments I see...
To estimate the cost of an item of equipment or plant from another of a known cost and size or capacity , we use a well known formula Cost B = Cost A * ( CapacityB/Capacity A)^n, where n is a factor usually around 0.5-0.8 depending on the plant or equipment involved. These exponents are...
Here is what I did :
work done in going from A to C,
W1 = 2nRToln(2) (isothermal process)
work done in going from C to B,
W1 = pΔV = nRΔT = -nRTo (isobaric process)
work done in going from B to A,
W3 = 0 (isochoric process)
so, total work done = W1 + W2 + W3...
Suppose on day-one a number is 15 then on day-twenty the number has increased to 200. Now I want to find out what that increasing number could be on day-forty by using the exponent derived from the day- one to day-twenty increase ; x(log15) = log 200 .
x = 2.301/1.176 = 1.956. So now on day...
I was just wondering what the proper notation would be to describe an infinite power tower which has each repeated exponent increasing by a value of 1, like such;
$$a^{{{{{{(a+1)}^{(a+2)}}^{(a+3)}}^{.}}^{.}}^{.}}$$
I am struggling to figure out how to calculate the expectation value because I am finding it hard to do something with the exponential. I tried using Euler's formula and some commutator relations, but I am always left with some term like ##\exp(L_z)## that I am not sure how to get rid of.
Two KL functions $f_1:\mathbb{R}^n\rightarrow \mathbb{R}$ and $f_2:\mathbb{R}^n\rightarrow \mathbb{R}$ are given which have KL exponent $\alpha_1$ and $\alpha_2$. What is the KL exponent of $f_1+f_2$?
In "The Theoretical Minimum" (the one on classical mechanics), on page 218, the authors write a Lagrangian
$$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$
They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be...
Homework Statement
Find the value of (-√3 + i)43/243
Homework EquationsThe Attempt at a Solution
I do not know how to really go about this problem.
I know that i0=1, i1=i, i2=-1, i3=-i, and I tried to use that to help but I got to no where, I also tried to break up the exponent into...
I've been taught that $$1^\infty$$ is undetermined case. Why is it so? Isn't $$1*1*1...=1$$ whatever times you would multiply it? So if you take a limit, say $$\lim_{n\to\infty} 1^n$$, doesn't it converge to 1? So why would the limit not exist?
<Moderator's note: Moved from a homework forum.>
1. Homework Statement
Given 0 < a < 1, i = √(-1),
ei2πa = cos 2πa + i sin 2πa
but also, ei2πa = (ei2π)a = 1a = 1
How to resolve the apparent contradiction?Homework Equations
eab = (ea)b
eix = cos x + i sin xThe Attempt at a Solution
No clue...
So this puzzle has been wandering the internet. I find it unsolvable because there is no way to determine what operation is being performed to the double fries. Everyone assumes it is addition, but addition and multiplication are performed with signs elsewhere in the puzzle. If we assume that...
$$3^x-14\cdot 3^{-x}=5$$
so expanded
$$x\ln{3}-\ln 7-\ln 2 +x \ln 3=\ln 5$$
ok W|A says the answer is
$$\frac{\ln\left({7}\right)}{\ln\left({3}\right)}$$
don't see the steps how?
I just asked a similar question, but I got help for that one, and now I am stumped again.
I need to find the domain for f(x) = ln(x^2-5x)
What's confusing me is how to deal with the exponent. I can't think of a way to get around it.
Please bare with me. Most of you know I actually don't have a great math background. In any case I'm going way back and filling in some very basic math that I have long forgot. I have some questions about terms in a polynomial.
Here is an example
$$3x^5+7x^3-5$$
1. From my book 3 and 7 are...
Hi, I have derived a matrix from a system of ODE, and the matrix looked pretty bad at first. Then recently, I tried the Gauss elimination, followed by the exponential application on the matrix (e^[A]) and after another Gauss elimination, it turned "down" to the Identity matrix. This is awfully...
Homework Statement
Solve the following coupled differential equations by finding the eigenvectors and eigenvalues of the matrix and using it to calculate the matrix exponent:
$$\frac{df}{dz}=i\delta f(z)+i\kappa b(z)$$
$$\frac{db}{dz}=-i\delta b(z)-i\kappa f(z)$$
In matrix form...
Homework Statement
Let's say I want to compute ##2^{2.4134}##. We know that the base is a rational number and the power is an irrational number. Please keep in mind that I have not taken too many math classes yet and I am self-studying right now by making a calculator and respective algorithms...
Homework Statement
Homework Equations
for equation which has 2 different solutions, D >0
The Attempt at a Solution
(1)[/B] D > 0
b^2 - 4ac > 0
3 - 4root2.k > 0
k < 3 / ( 4root 2 )
k < ( 3 root 2 ) /8
has solution of sin tetha and cos tetha
sin 0 = 0, cos 0 = 1.
when x = 0, and x = 1 -->...
Homework Statement
Evaluate each of the following expressions without using a calculator.
1) log216√8Solve for the unknown value in each of the following equations without using a calculator.
2) 3(x+4)−5(3x)=684
3) 7(42x)=28(4x)
Homework Equations
Exponent law for multiplication
The...
Hey folks, long time no see! :woot:
I am trying to estimate the Hurst Exponent for financial data using the R/σ (Rescaled Range) method. I have no authoritative source for this method. All I have are web pages (and we know how accurate those are). Specifically, I've been using the following...
The textbook proves that ##x^a x^b = x^{a+b}## by an induction argument on b. However, is an induction argument really necessary here? Can't we just look at the LHS and note that there are a ##a## x's multiplied by ##b## x's, so there must be ##a+b## x's?
I'm confused on this question.
The equation m log p (n) = q can be written in exponential form as..
The answer on the work sheet is p^(q/m)=n but shouldn't it be P^(qm) = n ? According to the power rule? My teacher explained this by writing down for me log p (n) = q / m but I'm confused here
Hey! :o
For each group $G$, $\text{exp}(G)$ is the exponent of the group $G$, i.e., the smallest positive integer $k$, such that $g^k=e$ for each $g\in G$.
Let $G$ be a finite group.
I have shown that $\text{exp}(G)$ divides $|G|$, and if $G$ is cyclic, then $\text{exp}(G)=|G|$, as follows...
Homework Statement
Sorry that I am not up on latex yet, but will describe the problem the best I can.
On the interval of a=1 to b= 4 for X. ∫√5/√x.
Homework EquationsThe Attempt at a Solution
My text indicates the answer is 2√5. I have taken my anti derivative and plugged in b and subtracted...
Homework Statement
Find critical numbers of the function: F(x)=t^3/4 - 2t^1/4
Derivative I got: F'(x)=3/4 t^-1/4 - 1/2 t^-3/4
Homework EquationsThe Attempt at a Solution
I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...
Hey guys,
I need your help. I am not sure if this is the right part of the forum to ask this question.
So I started reading papers about the Lyapunov Exponent, but there is something I do not understand in the formula. Why ? It seems logical that we want because we want to get the Exponent...
hello folks,
I am learning about how create a equation dimensional to formula Force (m*a).
I know this formula written this way:
[F]=[ma]
where:
m=mass
a=acceleration -> L/T²
The result is [F] = M * (L/T²) * L⁰
My result it's right?
Why T, relative time, has exponent two in formula...
Homework Statement
A distance R is measured to be 3.400±0.003m. What is the relative uncertainty in R? 9×10-4
What is the relative uncertainty in R^-2?
Homework EquationsThe Attempt at a Solution
Relative uncertainty of R is 0.003/3.4=8.8e-4
Relative uncertainty of R^-2= -2(8.8e-4)=-1.76e-3...
Homework Statement
How is ## e^log√(1-x^2)## equal to ##√(1-x^2)?##Homework EquationsThe Attempt at a Solution
taking ln on the function, ln√(1-x^2). lne⇒ ln√(1-x^2) ....
I work a lot in binary. I am organizing some of my work and need a way to write expressions. I can always create my own notation, but i would rather not invent something that already exists.
1011 is binary for 11 base 10. I use this {3,1,0} to represent the binary with just exponents. When I...
Hello all,
I am new to thermodynamics applied to turbines and compressors and I am trying to get my head around what is represented when calculating the work of a compression/expansion process using the polytropic exponent as oppose to the specific heat ratio of 1.4 (I'm working with air).
The...
Homework Statement
Let ##x \in G## and ##a,b \in \mathbb{Z}^+## Prove that ##x^{a+b} = x^a x^b##.
Homework EquationsThe Attempt at a Solution
If I am not mistaken, we would have to do multiple induction on ##a## and ##b## for the statement/proposition ##P(a,b) : x^{a+b} = x^a x^b##. First we...
Hello, folks. I'm trying to figure out how to take the partial derivative of something with a complex exponential, like
\frac{\partial}{\partial x} e^{i(\alpha x + \beta t)}
But I'm not really sure how to do so. I get that since I'm taking the partial w.r.t. x, I can treat t as a constant term...
Homework Statement
x(cnxn-1)
3. The Attempt at a Solution
I know that the answer is cnxn
I'm not sure why though. My thinking is that we have cnxn-1 and we multiply that by x. x times x is x2 so I'm expecting a 2 to interact with the n-1 in the exponents. I'm just not sure how n-1 interacts...
ok so is there a function that exists (for all intents and purposes let's call it G(x,y) )where
x= a^2*b^4*c
y=a^4*b^2*d
G(x,y) = a^2*b^4
basically gcd, but the exponents match those of the common prime factors of the first input (x)
********
equally useful would be a function where the...
Hi everyone,
I've encountered a curious problem I just can't figure out, and any input would be much appriciated!This is a personal project I'm working on, and as far as I know, there is no one else working on exactly the same. However, the computational study of critical phenomena is quite...