Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

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4
votes
7answers
612 views

Numbers to the Power of Zero

I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations. This is a three part question, the parts are as ...
1
vote
0answers
95 views

Can exponentiation and power function be defined through Albert Bennett's operations?

In 1914 Albert Bennett suggested the following operation: $$a * b=a^0_2b=\exp(\ln a \ln b)$$ Now, given this function, addition and multiplication, and their properties, can one express ...
0
votes
1answer
108 views

How do I solve this exponential equation?

$$x = 2^{x-3}$$ Does there exist an analytical solution to this equation? If so, how do I find it? What if it is changed to an equality? $$x>2^{x-3}$$
1
vote
4answers
1k views

Integration with infinity and exponential

How is $$\lim_{T\to\infty}\frac{1}T\int_{-T/2}^{T/2}e^{-2at}dt=\infty\;?$$ however my answer comes zero because putting limit in the expression, we get: $$\frac1\infty\left(-\frac1{2a}\right) ...
0
votes
1answer
126 views

Solution for an exponential expression without using logarithms, with two defined variables

If $60^a=3$ and $60^b=5$, what is the result of $12^{\frac{1-a-b}{-2-2b}}$? This has to be done without logarithms. The past four hours were helpless to me. Any hint, solution is welcome, I just ...
3
votes
1answer
158 views

Diophantine equation: fermat numbers and fibonacci numbers

My question is how to find all solutions $(m,n)\in\mathbb N^2$ for $F_n=f_m$, where $F_n=2^{2^n}+1$ and $f_m$ is the $m$th fibonacci number: $f_0=0$, $f_1=1$ and $f_n+f_{n+1}=f_{n+2}$ for each ...
1
vote
2answers
733 views

Solving system of recurrence relations

Base Case: $$ \left\{ \begin{array}{c} T(1) = 1 \\ T(2) = 1 \\T(3) = 4\end{array} \right. $$ I have the system: $$ \left\{ \begin{array}{c} T(N) = G(N-1) + F(N-1) \\ G(N) = F(N-1) + G(N-1) \\ ...
5
votes
7answers
471 views

Is $0^0=1$ postulate independent of all other axioms of complex numbers?

This question is inspired by the other question which asked for a proof that $i^i$ is a real number. Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
0
votes
2answers
1k views

Find the remainder of $128^{1000}/153$.

I have tried for almost more than an hour to find the remainder of the following $$\frac{128^{1000}}{153}$$ I applied remainder theorem to get the answer but could not succeed. Any suggestion or ...
2
votes
2answers
240 views

Find the greatest powers of $2$ dividing $10!$, $20!$, $30!$, $40!$ [duplicate]

I'm trying to find the greatest powers of $2$ dividing $10!$, $20!$, $30!$, $40!$, as part of a basic number systems course. I'm rather lost with this question. For $10!$ I tried writing the terms ...
7
votes
1answer
893 views

Why use radical notation instead of rational exponents?

I'm helping my younger sister for her math class. She has recently been taught integer exponents, and has starteed studying radicals (mainly square roots). The next topic will be rational exponents, ...
0
votes
2answers
77 views

Exponentation vs Power

What definition of $a^b$ operation is the most popular and standartized: Exponentation or Power? Is any difference between them?
4
votes
3answers
185 views

Solve $3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$

$$3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$$ I am completely lost on how to proceed. Could someone explain how to find any real solution to the above equation?
4
votes
2answers
996 views

upper bound of exponential function

I am looking for a tight upper bound of exponential function (or sum of exponential functions): $e^x<f(x)$ when $x<0$ or $\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$ Thanks a ...
1
vote
6answers
855 views

Identity proof $(x^{n}-y^{n})/(x-y) = \sum_{k=1}^{n} x^{n-k}y^{k-1}$

In a proof from a textbook they use the following identity (without proof): $(x^{n}-y^{n})/(x-y) = \sum_{k=1}^{n} x^{n-k}y^{k-1}$ Is there an easy way to prove the above? I suppose maybe an ...
8
votes
2answers
328 views

Is there another representation for $x^x$

I started wondering about this the other day. Since the following have their own alternate representations. $$\begin{align*} \displaystyle\large x+x=2x & \ \frac{x}{x}=1 & xx=x^2\end{align*}$$ ...
1
vote
1answer
54 views

Conventions for notation of function exponentation.

I read a previous question here but it seems incomplete for me (missing references). Given a generic function, $ f $ : 1. is true that $ f^2 $ means $ f^2(x) = (f \circ f)(x) = f(f(x)) $ ? 2. or is ...
1
vote
0answers
123 views

Solving for $x$ in $y=x^x(\ln x + 1)$ (Lambert W?)

I made a bunch of problems exercising the Lambert W-function in the solution, because I like to exercise to new concepts that I learn about. One that I came up with was rearranging $y = x^x(\ln x + ...
3
votes
1answer
226 views

Summation of powers inequality

Can anyone provide a slick proof of the following? Let $0 < x \le 1$. Then $\displaystyle \sum_{k=0}^{n-1} x^k \ge \frac {1} {1 - (1 - 1/n)x}$.
2
votes
0answers
472 views

exp(ab) decomposition

How can one write $e^{a(x) \cdot b(x)} = c(x) e^{ b(x) }$ with $c(x)$ not implicitly depending on $b(x)$. I do not believe this is generally possible so alternatively one can use an infinite series or ...
2
votes
2answers
59 views

regarding exponents, how to interpret and use.

Some times in the books both of the below mentioned concepts are used interchangeably. Is there any reason for that? When to use -(2)^2 = -4 and (-2)^2. Explain with any useful examples.
35
votes
11answers
3k views

How is $e^x$ read aloud?

My current research colleague from New Castle told me that I was reading it wrong. I usually read it as e power x. How do you read aloud $e ^ x$? Is it: e raised to x e power x e powered x or e ...
1
vote
2answers
132 views

Failure during calculating the matrix exponential, but where?

I have to calculate $e^{At}$ of the matrix $A$. We are learned to first compute $A^k$, by just computing $A$ for a few values of $k$, $k=\{0\ldots 4\}$, and then find a repetition. $A$ is defined as ...
1
vote
0answers
41 views

Triangular exponentation logarithm and inverse

The generalized formula of triangular exponentation on real numbers field is $x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)} $ It's my ...
6
votes
7answers
238 views

Solving $1.1^n = n^{100}$

How do I go about solving for $n$ in the following equation: $$(1.1)^n = n^{100}$$ A hint suffices.
4
votes
4answers
195 views

$(1+i)$ to the power $n$ [duplicate]

Possible Duplicate: Complex number: calculate $(1 + i)^n$. I came across a difficult problem which I would like to ask you about: Compute $ (1+i)^n $ for $ n \in \mathbb{Z}$ My ideas so ...
0
votes
1answer
138 views

Reducing the Index and Improper Fractions

I'm trying to do the problem 3√40x^4/y^9. When you try to reduce the index for 40^4, its going to be 4/3. How does the index get reduced into 2x√5x? I understand 3 cubed of 40, but what happens to ...
4
votes
2answers
107 views

Approximation with 1-exponential

How come that $$\left(1-\frac{1}{x}\right)^x \approx e^{-1}\ ?$$ Is there a proof or something to understand this?
6
votes
5answers
9k views

How do you calculate the modulo of a high-raised number?

I need some help with this problem: $$439^{233} \mod 713$$ I can't calculate $439^{223}$ since it's a very big number, there must be a way to do this. Thanks.
6
votes
1answer
251 views

Power of a block matrix with eigenvalues on the unit circle

In the expression $$\begin{bmatrix}A & C \\ 0 & B\end{bmatrix}^n = \begin{bmatrix}A^n & * \\ 0 & B^n\end{bmatrix},$$ I wonder whether the term denoted by * can be expressed in a simple ...
1
vote
2answers
38 views

How do I solve a function with x^2 and x^-1 to x?

We got two functions: $f(x)=ax^2+b$ $g(x)=x^{-1}=1/x$ I know that they are touching each other in $x=1$. Now I can find out the values for $a$ and $b$ in $f(x)$. Set the derivative of both ...
2
votes
1answer
223 views

Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?

Let $x$ be a positive real number. Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$. Call this function $f(n,x)$. Can we give good upper and lower bounds of $f(n,x)$ ...
1
vote
1answer
49 views

Problem understanding a proof about powers in ordered fields

I am reading through a textbook on Analysis and have come across a question that I can't seem to make any headway with. A proof is outlined, but I can't make any sense out of it. The problem is as ...
1
vote
1answer
607 views

Modulo of (Power of 2 divided by a number)

I wanted to calculate the power of $2$ raised to a number $a$, divided by another number $b$ and then take the modulo $K$ of this quantity. Meaning, I basically wanted $(2^a/b) \mod K$. Take an ...
0
votes
2answers
79 views

solving in x involving both exponential and logarithmic function

Is it possible to solve a function with both exponential and logarithm such as $a x^2−b.\log(x)= c$ in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?
1
vote
2answers
54 views

Elementary power equation: $k_1k_2^x = k_3k_4^x$

I have four constants $k_1$, $k_2$, $k_3$, $k_4$ and the following equation in an unknown $x$ (all are positive real): $k_1k_2^x = k_3k_4^x$ How do I solve for x?
6
votes
2answers
723 views

A “Matrix Trigonometry”

$e^X$ for matrix $X$ is defined as an always-converging taylor series (provided that $X$ is a $n \times n $ complex matrix): $$e^X:=\sum_{k=0}^{\infty}\frac{X^k}{k!} $$ A thought occurred to me that ...
1
vote
0answers
45 views

Is an equation of the following form solvable?

Is it possible to solve for $x$ which satisfies the equation $$d=(a\exp(bx)+\exp(cx))x^2$$ where $a,b,c,d$ are given constants? It looks quite horrible... Many thanks!
26
votes
1answer
363 views

Iterated exponent of $i$

WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
0
votes
3answers
392 views

Manipulating Exponents

I'm doing my homework and there are a couple of things that I am having trouble grasping. All my homework asks is that I simplify the exponents. For example: ...
6
votes
4answers
262 views

What is a nice way to compute $f(x) = x / (\exp(x) - 1)$?

I want it to be stable near $f(0) = 1$. Is there a nice function that does this already, like maybe a hyperbolic trig function or something like expm1, or should I just check if $x$ is near zero and ...
3
votes
1answer
436 views

Calculating the residue of power towers

I want to calculate the residue of a power tower. How do I do that? For example, I want to know the answer to this: $$2 \uparrow\uparrow 10 \pmod{10^9}$$
2
votes
1answer
187 views

Cardinal exponentiation problem from Halmos' Naive Set Theory

In chapter 24 of Halmos' Naive Set Theory the following problem is posed as an exercise (page 96): Prove that if $a, b$ and $c$ are cardinal numbers such that ${a}\le{b}$, then $a^c\le{b^c}$. ...
1
vote
0answers
99 views

How to solve these exponential equations for D?

I'm curious if this is even possible to solve for D. D is the only variable, x, y, z, and w are all constants, and e is the mathematical constant e. $$ (\frac{x+yD^{2}}{zD})^{\sqrt{2D}} = ...
1
vote
2answers
485 views

Writing a non integer power in terms of integer powers

I would like to write $x^{2.5}$ in terms of $x$ to the power of integers, is there any way to do this. Taylor series etc. don't work when they depend on derivatives. If it is not possible, do you ...
2
votes
3answers
160 views

How can I determine a formula for an exponential ratio?

I am not very experienced in mathematical notation, so please excuse some terminology misuse or formatting shortcomings. I have a project in which a value needs to increase from a set minimum to a ...
3
votes
2answers
2k views

How to handle big powers on big numbers e.g. $n^{915937897123891}$

I'm struggling with the way to calculate an expression like $n^{915937897123891}$ where $n$ could be really any number between 1 and the power itself. I'm trying to program (C#) this and therefor ...
3
votes
1answer
58 views

How is exponent notation related to this example

I am not sure what is meant by exponent notation and therefore how to answer this question is baffling me. Rewrite this in exponent notation: $\sqrt[3]{x^2y(z-X)^5}$
2
votes
1answer
218 views

Exponential function: Change base to exp

Why is the following true? $$\left[\frac{N - it}{N}\right]^{j+1} = \exp\left(-\frac{ijt}{N}\right)$$ i,j - integers less than N. Is there any theorem which allows me to get this result? I tried ...
8
votes
4answers
379 views

Is there a Definite Integral Representation for $n^n$?

The factorial $n!$ has a nice representation as definite integral: $$ n!=\Gamma(n+1)=\int_0^\infty t^{n} e^{-t}\, \mathrm{d}t. \! $$ Is it possible to write down such an integral for $n^n$ as ...