Questions about exponentiation

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0
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2answers
937 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
181 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
675 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
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3answers
182 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
575 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
703 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
321 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
50 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
120 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
185 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
420 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
58 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
122 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
229 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
181 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
122 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
104 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
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5answers
7k 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
220 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 ...
-4
votes
1answer
117 views

Boundedness on strips in the complex plane for functional equations [closed]

We know that the recurrence for $b>0$ (1) $f(0)=1$ (2) $f(z+1)=b{f(z)}$ has $f(z)=b^z$ as the only entire solution that is bounded on the strip $S=\{z: 0<\Re(z)\le 1\}$. The image of $S$ ...
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
213 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
558 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
74 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
638 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
44 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!
23
votes
1answer
349 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
309 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
258 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
366 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
178 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
97 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
422 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
141 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
1k 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
189 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
373 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 ...
1
vote
2answers
133 views

Cycling through powers of a generator of finite field. Something similar to modpow for Z/nZ

I am not sure if I am talking to the correct community (perhaps stack overflow is best). I want to be able to compute $$g^x = \underbrace{g \cdot g \cdot \dots \cdot g}_\text{x amount of times}$$ ...
0
votes
3answers
526 views

How complex exponential converges and “sum of exponents” rule holds

How is it the complex exponential converges for any value of $z$ in the complex plane? $$e^{z} = 1 + \frac{z}{1!} + \frac{z^2}{2!} \cdots\cdots$$ How is it the "sum of exponents" rule holds for ...
0
votes
2answers
96 views

About Square Rooting

I have read that "every positive number $a$ has two square roots, positive and negative". For that reason I have always (as far as I could remember) unconsciously done the following for such ...
0
votes
2answers
151 views

Why does $(10^4 - 10^2) \cdot 0.0012121212\dots = 12$?

When you answer this question $(10^4 - 10^2) \cdot 0.0012121212\dots$ you get $12$. However, that seems to defy PEMDAS. Please explain. Doing PEMDAS wouldn't you get $(10^4 - 10^2)$ = $10^2$ and then ...
4
votes
1answer
277 views

How to solve infinite repeating exponents

How do you approach a problem like (solve for $x$): $$x^{x^{x^{x^{...}}}}=2$$ Also, I have no idea what to tag this as. Thanks for any help.
0
votes
1answer
79 views

Exponential Growth

A bacteria culture is known to grow at a rate proportional to the amount present.After $1$ hour $1000$s stands of the bacteria are observed in the culture;and after $4 $years $3000$ strands. Find: ...
12
votes
2answers
548 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...