Questions about exponentiation

learn more… | top users | synonyms (1)

1
vote
2answers
46 views

Writing $e^{i\theta}(e^{in\theta}-1)/(e^{i\theta}-1)$ in $(a+i b)$ form

How to write: $$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}$$ in $$(a+i b) $$ $$ ?$$ I tried to multiplicate by $$e^{i}$$ (the numerator and ...
3
votes
1answer
42 views

Infinite Perfect power of numbers in a certain form

A question I found very interesting , which I found written on a blackboard while visiting a near by community science center is as follows. Prove that there exist infinitely many $m,n,k$ for ...
1
vote
1answer
34 views

Lower bound for $(x + y)^k $?

I'm wondering, is there a lower bound for $(x + y)^k $? For example, if $x,y,k > 0$, can we say that $(x + y)^k \geq x^k + y^k$? If anyone has a source/reference for this, that would be great.
0
votes
1answer
12 views

Finding an exponential formula passed upon the start and end points.

I'd like to create pricing curve that's based upon a reverse exponential function. I know the starting point and ending point, but don't know how to create the curve in between. For example, say for ...
0
votes
4answers
65 views

If $10^{80}=2^x$, what is the value of $x$?

If $10^{80}=2^x$, what is the value of $x$? (Or, what binary word length would you need to contain $10$ to the $80$?)
5
votes
1answer
146 views

Solving the equation $a ^ b + b ^ a = 200$

Find $a$ and $b$, $a ^ b + b ^ a = 200$ One of the answers is $a = 1$ and $b = 199$. Lets say $a, b$ belongs to $\mathbb{R}$ then there will be many solutions, for each $a$ there exist $b$, in ...
1
vote
1answer
21 views

Find highest power of 2 that divides $3^{2^k}-1$

I am trying to find highest power of 2 that divides $3^{2^k}-1$ but I have no idea where to start - could you give me any hint?
0
votes
2answers
25 views

Exponent of Projection (T^2=T)

T:V->V is a linear transformation, also it's a projection, i.e. T^2=T. Find e^T. I thought of using the fact that if T=T^2 then e^T=e^(T^2) but I guess that doesn't work because exponent is a sum of ...
0
votes
1answer
19 views

Exponent of polynomials (of matrices)

$A$ is a matrix over $\mathbb R$ (reals). Prove that for every $f,g\in \mathbb R[x]$, $\displaystyle e^{f(A)}\times e^{g(A)} = e^{f(A)+g(A)}$ I tried using the sigma writing but got stuck (I ...
0
votes
1answer
44 views

Methods for solving equations with exponents?

In the following equation, capital letters represent arbitrary real numbers that are constant with respect to $x$: $$A\left(x+B\right)\left(1 + \frac{C}{x+D}\right)^E + Fx + G = 0$$ I'm trying to ...
4
votes
2answers
114 views

An “elementary” approach to complex exponents?

Is there any way to extend the elementary definition of powers to the case of complex numbers? By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots ...
1
vote
1answer
32 views

Fractional index fallacy

According to index laws, $(a^b)^c=a^{b\cdot c}=(a^c)^b$ However, if for example we have $a=-1, b=4, c=1/2$, then we get the equation: $$((-1)^4)^{1/2}=(-1)^2=((-1)^{1/2})^4$$ The first equation is ...
0
votes
1answer
65 views

Is there any easy way to find the positive integer solutions $(x,y,z)$ from this linear equation?

The equation is like this: $3^x -2^y = 19^z$ It seems that no way to find the solution except using trial and error. I got only one solution: $x=3, y=3, and z= 1$ by using trial and error. But, ...
0
votes
0answers
22 views

Proof of equations of exponentials

I am working on a proof of exponential equations. Before this, I have verified the correctness, or approximate correctness of the equations by numerical results. The questions is: Given: 1) ...
1
vote
3answers
66 views

How to prove that $\frac{a^n}{a^m}$ is equal to $a^{n-m}$? [closed]

How to prove that $\dfrac{a^n}{a^m}$ is equal to $a^{n-m}$? Thank you in advance.
0
votes
1answer
29 views

Dividing 1 by powers of ten

I have something like, say, 3.9 x 10^-22 and I want it to be divided by 1. This is common in Chemistry science questions, but I fail to get the thing. How can we divide 1 by 3.9 x 10^-22 and get a ...
0
votes
1answer
41 views

Reference Request: Nicole Oresme history

It says on Wikipedia that [Nicole Oresme] also worked on fractional powers, and the notion of probability over infinite sequences, ideas which would not be further developed for the next three and ...
2
votes
3answers
156 views

Find positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$

Find all positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$ I encountered this question in one of my monthly assignments. Unfortunately, I don't know ...
6
votes
3answers
129 views

Primes as a difference of powers

Find the smallest prime that cannot be written as $$|3^a - 2^b|$$ EDIT: I forgot to mention that $a$ and $b$ are whole numbers. I tried to expand $3^a$ as $(2+1)^a$ using binomial theorem but ...
2
votes
1answer
24 views

How to simplify this expression that contains exponential terms?

In a multiple choice exam , I encountered the following question. The answer to the question is $$ \frac{17}{8}.$$ The question is: $$\frac{16^{x+1}+4^{2x}}{2^{x-3}8^{x+2}} \text{ is ? }$$
0
votes
1answer
42 views

Exponential function to prove [closed]

how would you prove that $Ae^x+Be^{-x}=A \sinh x+B\cosh x$ Thank you.
0
votes
1answer
37 views

Describing the exponantiation of a number by itself

When a mathematician says What is the square of $n$? It is generally understood that the expected answer would be to multiply $n$ by itself, $n^2$. Is there a word analogous to square to ...
1
vote
1answer
39 views

Can someone look at my proof about the convergence of $e^{-tA}$

Hi I am trying to prove that if A is a symmetric positive definite matrix then $e^{-tA}\rightarrow 0$ as $t\rightarrow\infty$. So I have attempted an answer but I'm not sure it is correct. ...
1
vote
1answer
32 views

How fast will a shape grow if it can grow exponentially only at the border, and growth is limited by crowding?

Take a hypothetical bacterium which divide once per minute. After $n$ minutes there will be $2^n$ bacteria, assuming no constraints. But what if its growth is constrained by resources and space? I am ...
0
votes
0answers
27 views

Finding of Exp(Jt), there J - Jordan form

I need to solve this: x(t)=exp(At)x0 To find exp(At) i need find exp(Jt), J(A) - Jordan form of A; ...
0
votes
0answers
36 views

Quick question on matrix exponentiation.

Hi can someone explain a step in the proof that if A, B are commutative matrices then $e^Ae^B=e^{A+B}$ So define $f(t)=e^{tA}e^{tB}$ then $f(0)=I$ and we have that $f'(t)=Ae^{tA}e^{tB}+e^{tA}Be^{tB}$ ...
0
votes
1answer
31 views

Why is $\left(1-\frac{1}{k}\right)^t < e^{-t/k}$?

I came across this statement, but can't see why it holds: $\left(1-\frac{1}{k}\right)^t < e^{-t/k}$ I'm sure it's something simple, but I don't have a great deal of mathematical experience. I ...
0
votes
1answer
61 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
4
votes
3answers
358 views

Solving exponential equations using logarithms

This is the equation that I am having troubles with: $$\large x^{\large\log_{10}5}+5^{\large\log_{10}x}=50$$ So the first thing I do, I logarithm the whole expression with $\log_{10}$. So I get: ...
0
votes
0answers
32 views

$n^0=1$ and $0^n=0$, so what is $0^0$? [duplicate]

Today I stumbled accross this mathematical dilemma: $0^0$. There are two simple rules in mathematics regarding the expression of potency. $$n^0=1$$ $$0^n=0$$ Where $n$ is any number. But this leaves ...
0
votes
2answers
90 views

Is there a way to write $2^3+2^2+2^1+2^0$ in short form or a better way?

I am doing a question and instead of going through phases solving the question I was wondering if I could do it all in one with a short equation. The question is about compound interest finding the ...
1
vote
1answer
71 views

Proof that irrational exponents exist, and are unique?

Can someone provide a proof that for any given irrational number, $b$, exponentiation by that number defined as a limit of rational powers always converges, and that if we choose a particular base, ...
1
vote
0answers
26 views

Imaginary Exponentiation, and Exponentiation in General [duplicate]

I think I may have asked a similar question before, but first-how does one go about computing imaginary exponentials? Secondly, what is the formal definition of an irrational (or I guess more ...
0
votes
3answers
41 views

Is there a way to find x when x is an exponent?

I'm kind of stuck on how to solve the following. $10^x = 5^9$ Is there a method or a simple trick to find what is x?
1
vote
1answer
40 views

Logarithmic Contest Question

The Problem was as follows: Define $\log*(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to $1$. For example ...
1
vote
1answer
46 views

Power Factoring Contest Question

The question was as follows: Compute the smallest positive integer $n$ such that $n^n$ has at least $1,000,000$ positive divisors. I did some work, finding that if $n=2^a*3^b*5^c*7^d$ then the $n^n= ...
0
votes
1answer
29 views

Exponential percentage decrease based on time

I have a bar that shows the time left for a task to finish and I want it to decrease faster as it gets closer to the end time. Example: Let's assume that the total time required for Task A to ...
0
votes
1answer
40 views

How can I derive this summation?

I have the following equation, $$ K_r=\left ( \frac{P}{RT} \right )^{v}exp \left \{ \sum_{s}\left [ (\beta_{s,r}-\alpha_{s,r}) \left \langle \frac{h_s}{RT}-\frac{s_s}{R}\right \rangle \right ] ...
0
votes
2answers
48 views

Convergence of a series ${}\qquad{}$

Does this series converge? $$\sum_{x=2}^n \left(\frac{1}{x}\right)^{\left(\frac{1}{x}\right)}$$ I tried hard but stil had problems... Could someone help me?
1
vote
1answer
26 views

Could somebody validate my proof for the limit of $a^{x_n}$ when $x_n \to c$?

So, here is the clear formulation of the problem: let $(x_n) $ be a convergent sequence of positive numbers, with $x_n \to c$. I want to prove that the sequence $(y_n) $, with $y_n=a^{x_n} $, ...
0
votes
0answers
17 views

How we can calculate the power of an interval?

We know if two intervals are uncorrelated like $X=[a,b], \; Y=[c,d]$ the product of $X$ and $Y$ is: $X \times Y = [\min(S),\max(S)], \; S = (ac,ad,bc,bd)$ But for powers, if the intervals are ...
1
vote
2answers
188 views

Properties of Logarithms

How do you simplify the following expression? $$\log\left(3^{(5^7)}\right)$$ I know that logarithms are like the inverse of exponents, but are there any tricks to simplify powers inside logarithms? ...
0
votes
0answers
104 views

Integrating quotients with polynomials in numerator and denominator that are raised to fractional powers

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
0
votes
0answers
31 views

How to calculate sum of digit of a power

Find the sum of the digits of: $$\left\lfloor\frac{k^{h+1}-1}{h-1}\right\rfloor$$ I need to calculate sum of digits in answer. Note as $k$ and $h$ can be a very big value, answer is getting ...
0
votes
1answer
25 views

Economics question. Rehashing the basics of dealing with exponents

Struggling to put two equations together effectively again. I have my income equation: $$ y=zk^\alpha $$ And I'm trying to plug it into my marginal product of capital $$ MPK=\alpha z k^{\alpha-1} ...
0
votes
4answers
71 views

What is the name of the answer to exponentiation?

What is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to ...
0
votes
2answers
31 views

Simplifying $\exp {- i 2 \pi / N}$.

a and b are complex numbers and I know the equation below. $$X_{N} = a + e^{-i2\pi /N}*b$$ I wanted to simplify it. Here is what I've tried. I know $e^{-i\pi} = -1$ $X_{N} = a + \left ( e^{i\pi} ...
0
votes
2answers
21 views

Exponents with variables inside exponents

I am confused about how to reduce this, is there any way? $\sqrt x \ln(\sqrt x) = ln(\sqrt x^\sqrt x) = $ $?$ This can be written like this too: $ln(x^\frac{x^\frac{1}{2}}{2}$) Or: ...
0
votes
0answers
12 views

Find the value of x if $\log_{2}({5 * 2^x + 1})$, $\log_{2}({2^{1 -x} + 1})$ and 1 are in arithmetic progression

My try: $1 - \log_{2}({2^{1 -x} + 1})$ = $ \log_{2}({2^{1 -x} + 1}) - \log_{2}({5 * 2^x + 1})$ $10 * 2^x + 2 = (2^{1 -x} + 1 )^2$ Let $2^x = t $ $10t^3 + t^2 -4t -4 = 0$ But I ...
0
votes
2answers
24 views

Non-integer exponents of negative numbers?

There is a formula for exponents of negative numbers as follows: $m^n=(-1)^n|m|^n$. This formulation works when $m<0$ and $n\in \mathbb{Z}$. But what about for $n\in \mathbb{R}$? Is there a ...