Questions tagged [exponentiation]

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

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4
votes
3answers
107 views

EGMO 2014/P3 : Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist ...
9
votes
2answers
131 views

why $\left(\left( \left(-\frac{1}{4}\right)^{-2}\right)^\frac{1}{4}\right) \neq \left(\left(-\frac{1}{4}\right)^{-\frac{1}{2}}\right)$?

Why $\left(\left( \left(-\frac{1}{4}\right)^{-2}\right)^\frac{1}{4}\right) \neq \left(\left(-\frac{1}{4}\right)^{-\frac{1}{2}}\right)$? well, I tried this question but as far my calculations, I am ...
-1
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1answer
65 views

Continuity of $f(t)=t^{x-1}$

In our material we have the statement that for a given $x\in\mathbb{R}$ with $x\geq 1$ the function $f(t):[0,1]\to \mathbb{R}$, where $f(t)=t^{x-1}$, is continuous (if we define $0^0:=1$). Then a few ...
2
votes
1answer
66 views

Can we find a necessary and sufficient condition to have : $(x^2)^{1-f(x)}+((1-x)^2)^{1-f(1-x)}\leq 1$?

It's a possible generalization of an inequality of Vasile Cirtoaje I recall : Let $x,y>0$ such that $x+y=1$ then we have : $$x^{2y}+y^{2x}\leq 1$$ Well the idea is really simple take a function $...
4
votes
0answers
93 views

Are powers of a natural number distributed widely in the residue system?

I encountered the following while solving another problem and need some help. Let $p,d$ be relatively prime natural numbers. I think that for many of $(p,d)$ tuples, numbers in set $\mathbb{N}_d = \{0,...
2
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2answers
30 views

Interpretations of Exponents

I have been reading one of the proofs of Euler's identity, $e^{i\theta}=\cos(\theta)+i\sin(\theta)$. I have always thought that exponents can be interpreted as its base being multiplied its exponent ...
0
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1answer
66 views

Solution to : $g(x)=(\sin^2(x))^{\frac{1}{f(x)}}+(\cos^2(x))^{f(x)}\leq 1$

Well, this is related to my own question Pretty conjecture $x^{\left(\frac{y}{x}\right)^n}+y^{\left(\frac{x}{y}\right)^n}\leq 1$ . Let $x\in(-\infty,\infty)$ and define $g(x),f(x)$ continuous and ...
2
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1answer
66 views

How do I find the value of $\operatorname{LCM} (a_1^k,a_2^k, \ldots ,a_n^k) \pmod {m}$?

LCM can be upto or greater than $10^{50}$. $k$ and $m$ are upto $10^9$. My current approach is Finding the LCM . (LCM % m)^k=A. A % m. This is working but takes a long time in finding LCM using GCD. ...
3
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3answers
286 views

How to prove that $(a^m)^n=a^{mn}$ where $a,m,n$ are real numbers and a>0?

I know how to prove the equality when $m$ is a rational number and $n$ is an integer, but do not know how to go about proving this for real numbers. On a semi-related note, I was trying to prove this ...
0
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0answers
25 views

Is there a name for this softmax-like equation?

I am working on some software and have a need to normalize a list of values. I originally was using a typical softmax, but found that it had some undesirable properties for what I was doing. I wound ...
3
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0answers
88 views

What does the exponential of a vector do geometrically?

The exponential of an even multi-vector is related to rotation, but what is the exponential of a vector? For instance, the exponential of a vector $\mathbf{v}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{...
0
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3answers
131 views

why are $e^{2x}$ and $e^{x^2}$ inequal? [closed]

enter image description here From the index rules I learned from school, $a^{x^2}=a^{2x}$ Does it work the same for the natural constant? Why is it?
1
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0answers
128 views

Conjecture $a^{(\frac{a}{b})^p}+b^{(\frac{b}{a})^p}+c\geq 1$

At the beginning I would like to create a question about how I find my inequalities but it seems more complicated than obvious so I propose an inequality of mine which is also a conjecture : Let $a\...
0
votes
2answers
26 views

Exponential growth and decay,and the constant k

In the equation $$A=Be^{kt}$$,where $B$ is the initial amount and $t$ is the time taken what is $k$,I know it's a constant of proportionality ,but is it the same as the number of time a certain amount ...
9
votes
1answer
305 views

Pretty conjecture $x^{\left(\frac{y}{x}\right)^n}+y^{\left(\frac{x}{y}\right)^n}\leq 1$

inspired (again) by an inequality of Vasile Cirtoaje I propose my own conjecture : Let $x,y>0$ such that $x+y=1$ and $n\geq 1$ a natural number then we have : $$x^{\left(\frac{y}{x}\right)^n}+y^{\...
1
vote
1answer
57 views

Is there a name for the function that gives you the base?

We can take a base and raise it to an exponent to get a result, e.g. $2^5 = 32$. This is basically from a power or exponential function, which fixes the exponent or base and takes as input the other, ...
2
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0answers
64 views

How do I solve this equation, which includes logarithms,

So I’m having difficulties with an equation involving a $\log(x)$. I was actually trying to solve a physics problem, but since this is an issue relating maths I came here. Anyway, my equation is: $-9 =...
4
votes
2answers
47 views

Rates of change, compounding rates and exponentiation

I have a very (apologies if stupidly) simple question about rates of change that has been bugging me for some time. I can't work out whether it relates to my misunderstanding what a rate of change is, ...
-1
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1answer
30 views

Simple cardinal arithmetic

How can I see that $$2^{2^\lambda}>2^\lambda$$ ? Is it used here that $\lambda \geq 2^{\aleph_0}$ ? The reference is here, pages 4 and 8.
0
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4answers
72 views

Why is $\sqrt{x^2}$ is $|x|$? [closed]

I was trying a problem and was getting the wrong answer and when I saw the solution on the internet I found this statement written in square brackets $\sqrt{x^2}$[note square is on $x$] is $|x|$. Till ...
6
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1answer
57 views

Set of numbers with interesting property

There is an interesting property with the set $(2,2)$ $$2 + 2 = 4$$ $$2\times2 = 4$$ $$2^2 = 4$$ I am wondering is (2,2) the only set with this property. For example, consider the set $(1,2,3)$ $$1+2+...
4
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0answers
83 views

$x^n+y^n+z^n+u^n+v^n=(x+y+z+u+v)^{n-1}, 3 \le n \le 5$

$$\left\{\begin{eqnarray*}{} x^5+y^5+z^5+u^5+v^5&=&(x+y+z+u+v)^4\\ x^4+y^4+z^4+u^4+v^4&=&(x+y+z+u+v)^3\\ x^3+y^3+z^3+u^3+v^3&=&(x+y+z+u+v)^2\\ xyzuv&=&1 \end{eqnarray*}\...
3
votes
1answer
93 views

Solve $x^{x^{x^{2017}}}=2017$

I have tried to use $\ln$, but couldn't solve: \begin{equation} \ln x^{x^{x^{2017}}}=x^{x^{2017}}\ln x=\ln 2017. \end{equation} I found that $x=\sqrt[2017]{2017}$ is a solution, and it is easy to ...
0
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0answers
60 views

When is $(a^b)^c$ not equal to $a^{bc}$?

In this dsp.SE answer, the subtle point that $e^{-j\omega t} \ne ({e^{j\omega}})^{-t}$ is made. At school I learned that $(a^b)^c = a^{bc}$, which is also supported by answers like this one. When does ...
3
votes
1answer
53 views

$x^{5x}=y^y$, $x, y \in \mathbb{Z}^+$, find largest value of $x$. [closed]

Let $x$ and $y$ be positive integers satisfying $x^{5x} = y^y$. What is the largest possible value for $x$? I'm stuck on this question in an Olympiad past paper. Anyone have any ideas about this one?
1
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1answer
79 views

Converting recurrence relation to linear and solve with matrix exponentiation

Let's say we have the recurrence relation $$ x_n = \begin{cases} x_{n-1} + y_{n-2} + y_{n-3} + n2^n, & \mbox{if } n\ \geq 0 \\ 1 & \mbox{if } n \lt 0 \end{cases}\\ y_n = \begin{cases} y_{n-2} +...
0
votes
1answer
67 views

Prove that there are infinitely many primes of the form $2^n-n^2$

My original question is: Find all the number $x$ such that, for all $n \in \mathbb{N}, n\ge 5$, we have $ 2^n-n^2 | x^n -n^x$. The first thing to notice is that, if $n$ is even, then $2^n-n^2$ is even ...
0
votes
2answers
35 views

I don't understand this exponent simplification

I've been doing the Khan Academy math courses to brush up on my math foundations before starting my CS/math degree in the fall semester. I just don't fully understand negative exponents, I stumbled ...
1
vote
0answers
46 views

Is there a term for the function $\operatorname{sign}(x) \times |x|^a$?

I wanted to use a power function for some procedural generation, but also needed to preserve the sign of the input and not be limited to uneven integer exponents. This seems like a basic enough use ...
0
votes
0answers
13 views

Exponential handling

I saw this equation somewhere: $ e^(- Ef/kt)$ = $ e^(- Ev1/kt)$ + 0.5$ e^(- Ev2/kt)$. The author then ends to this: $ e^(- Ef)$ = $ e^(- Ev1)$ + 0.5$ e^(- Ev2)$. All the kt are removed. I did not ...
5
votes
1answer
146 views

Nice inequality with exponents $a^{2b}+b^{2a}\leq a^{\Big(\frac{a(1-a)(\frac{1}{2}-a)}{4}\Big)^2}$

Hi it's a little refinement to play with a hard inequality of Vasile Cirtoaje : Let $a\geq b>0$ such that $a+b=1$ then we have : $$a^{2b}+b^{2a}\leq a^{\Big(\frac{a(1-a)(\frac{1}{2}-a)}{4}\Big)^2}...
0
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0answers
13 views

For two consecutive unequal periods of time and the number of events in each, can you construct a growth formula that stays accurate for both periods?

For example: Period #1 is 4 days long, and had 5 events occur Period #2 is 3 days long, and had 6 events occur Is it possible to construct a formula that projects the growth of the occurrences of ...
1
vote
0answers
51 views

if given $3^{6x}+3^{-6x}+3^{2x+1}+3^{-2x+2} = 27 $, then the value of $ \frac{2(a+b)}{3^{a} + 3^{b}}$ where a and b is roots of the given.

I never learned this in class. Given: $$3^{6x}+3^{-6x}+3^{2x+1}+3^{-2x+2} = 27 $$ If a and b are the solutions, then the value of: $$ \frac{2(a+b)}{3^{a} + 3^{b}}$$ are ....? I tried to let $p=3^{2x}$,...
4
votes
1answer
90 views

Arrange irrationals in ascending order:$ 2^{\sqrt{\frac{5}{3}}},3^{\sqrt{\frac{3}{5}}},5^{\sqrt{\frac{4}{15}}},29^{\frac{1}{\sqrt{15}}} $

How to arrange following irrational numbers in ascending order: $$ 2^{\sqrt{\frac{5}{3}}}, 3^{\sqrt{\frac{3}{5}}}, 5^{\sqrt{\frac{4}{15}}}, 29^{\frac{1}{\sqrt{15}}} $$ My try: Using a calculator I ...
1
vote
2answers
99 views

Finding a closed form to a minimum of a function

It's a try to find a closed form to the minimum of the function : Let $0<x<1$ then define : $$g(x)=x^{2(1-x)}+(1-x)^{2x}$$ Denotes $x_0$ the abscissa of the minimum . Miraculously using Slater's ...
-2
votes
1answer
43 views

What is the fundamental error in my reasoning?

What is fundamentally wrong in writing $(-a)^{1/2}$ as $((-a)^{2})^{1/4}$ when $a$ is positive and thus equating it to $a^{1/2}$? Edit: I'm basically asking if there is anything wrong with this ...
0
votes
4answers
97 views

Explain and Apply Euler's Generalisation of Fermat's Theorem

As I understand Euler's Generalization of Fermat's little theorem in Modulo Arithmetic, it is: $a^{\phi(n)} \equiv 1 \pmod{n}$ However, I have also seen a version of the theorem which seems more ...
2
votes
2answers
44 views

Solving $2^n - 2\times n = a $, where $a$ is a known constant

Solving $2^n - 2\times n = a $, where $a$ is a known constant. This is my first question. I am having trouble solving the equation in the title...moreover I do not even know the name of that kind of ...
1
vote
5answers
94 views

What is wrong with $x=x^{2/2}=\sqrt {x^2}=\lvert x\rvert$

The title says it all, $x=x^{2/2}=\sqrt {x^2}=\lvert x\rvert$ cannot possibly be true, so what am I missing?
1
vote
3answers
32 views

What maths rule allows this expression with powers to be rewritten as below?

I have been reading through a programming book and the author asked to calculate the following expression in the program. $$4*5^3+6*5^2+7*5+8$$ I approached this by expanding the expression like so: $$...
1
vote
0answers
156 views

Solve : $\Bigg(\frac{ab}{a^{2a}+b^{2b}}\Bigg)^{a^n2^{n-1}}+\Bigg(\frac{ab}{a^{2a}+b^{2b}}\Bigg)^{b^n2^{n-1}}\leq 1 $

Let $a,b>0$ such that $a+b=1$ and $n\geq 2$ a natural number then we have : $$\Bigg(\frac{ab}{a^{2a}+b^{2b}}\Bigg)^{a^n2^{n-1}}+\Bigg(\frac{ab}{a^{2a}+b^{2b}}\Bigg)^{b^n2^{n-1}}\leq 1 $$ We have ...
0
votes
0answers
33 views

Irrational Number modulo Prime

Given three positive integers $a$,$r$ and $n$, I need to calculate $\left\lfloor(a+\sqrt{r})^n\right\rfloor$ mod $p$, where $p$ is a prime. Let $\left\lfloor(a+\sqrt{r})^n\right\rfloor=c+d\sqrt{r}$. ...
0
votes
1answer
66 views

Proving exponent law for real numbers using the supremum definition only

I was working on a problem displaying the expansion of the definition of exponents, and naturally the final question was to prove the exponent laws when the exponents are real numbers. For $a>1$, ...
0
votes
0answers
23 views

How does this differential equation's solutions embedded in an exponential-exponential coordinate plane change the form of the differential equation?

$u=\log(x)$ and $v=\log(y)\implies uv=k.$ $uv=k$ satisfies an algebraic differential equation. $X=(x,-y)$ is a vector field with integral curves $uv=k$ in $u-v$ coordinates. A differential equation is ...
0
votes
1answer
47 views

Inverting the function $f(x)=x\cdot C^x$

Let $C>0$ be a positive constant. Then consider the function $$ f(x)=x\cdot C^x.\qquad(x>0) $$ How do I compute $f^{-1}(y)$? So far, I have only been able to derive $$ \begin{align*} y=x\cdot C^...
6
votes
5answers
615 views

Is there a relation between fractional power and logarithm functions?

Something that has always bothered me is that there's no way to get $x^{-1}$ by differentiating $x^a$ for some $a$, even though all other negative powers of $x$ can be achieved by differentiating some ...
1
vote
1answer
59 views

Fast modular exponentiation for $60^{53} \text{ mod } 299$

I'm trying to find the modular exponentiation for $60^{53} \text{ mod } 299$. I know it is $21$, but I would like to to show the answer step by step so that a normal calculator (with no modulo ...
0
votes
1answer
58 views

how to define $x^\sqrt 2$? [duplicate]

Can anyone please tell me how to define $x^\sqrt 2$ ? If it was $x^2$ or $x ^ \frac{1}{2}$, we could have said that $x^2$ means $x \times x$ and $x ^ \frac{1}{2}$ means a number y such that $y^2 = x$....
2
votes
2answers
63 views

Exponent notation (Tetration)

What is the meaning of this notation, and how does it work? $$\exp_{10}^2(1.09902),\,\exp_{10}^3(1.09902)$$ I knew this notation when i was reading about tetration on wikipedia. Here is the link: ...
0
votes
3answers
41 views

If $x \in \mathbb{Q}$ and $b \in \mathbb{R}$ with $b > 1$ and $x > 0$, then $b^x \geq 1$ [closed]

The following problem is motivated from Baby Rudin, Chapter 1, Problem 6(c): if $x \in \mathbb{Q}$ with $b > 1$ and $x \geq 0$, then $b^x \geq 1$. I would prefer not to use techniques such as ...

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