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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15
votes
0answers
216 views

If $r\in\mathbb{Q}\setminus\mathbb{Z}$ is it possible that $r^{r^{r^r}}\in \mathbb{Q}$?

It's straightforward to prove that $r^r\notin\mathbb{Q}$, which furthermore allows us to use the Gelfond-Schneider theorem to prove that $r^{r^r}\notin\mathbb{Q}$. Is it true that $r^{r^{r^r}}\notin\...
14
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0answers
588 views

When does $x^{x^{x^{…^x}}}$ diverge but $x^{x^{x^{…^c}}}$ converge?

Let us define these two sequences as follows: $a_0=1$, $b_0=c$ $a_{n+1}=x^{a_n}$, $b_{n+1}=x^{b_n}$ $b_{n+1}\ne b_n$ for any $n$. $x,c\in\mathbb C$ Is it possible for $a_n$ to diverge but $b_n$ ...
13
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1answer
208 views

Is there any simple ways to compare $x^y$ and $y^x$ without a calculator?

There are plenty of discussion on MSE about how to compare $x^y$ and $y^x$. For $x,y>e$, it is sufficient to just compare $x$ and $y$ to reach a conclusion. But I wonder if there are some general ...
11
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0answers
199 views

When does $2^n$ start with n?

A trick my dad taught me for easily referencing powers of 2 is that $2^6=64$ and $2^{10}=1024$, because $64$ starts with $6$ and $1024$ starts with $10$, and so it's faster than manually doubling ...
9
votes
1answer
316 views

Expressing the values of a matrix at pow N

I have a square matrix (that comes from a Markov Chain) that looks like that: $$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 ...
7
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0answers
301 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
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+...
6
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0answers
186 views

Way to express a number in its most compact sum of powers

Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...
6
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0answers
184 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate $\...
6
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0answers
339 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
6
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0answers
370 views

Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} (...
6
votes
1answer
143 views

Prove that if $2^x,3^x, 5^x, 7^x, 11^x … $ are all integers then $x$ is an integer as well

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
5
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0answers
266 views

Monstruous inequality $\Big(a^{(4b^2)^{4a^2}}+b^{(4a^2)^{4b^2}}\Big)^{\frac{1}{2}}\Big(a^{(4b^2)^{2b}}+b^{(4a^2)^{2a}}\Big)^{\frac{1}{2}}\geq 1$

Hi I want to solve this , Let $a,b>0$ such that $a+b=1$ then we have : $$\Big(a^{(4b^2)^{4a^2}}+b^{(4a^2)^{4b^2}}\Big)^{\frac{1}{2}}\Big(a^{(4b^2)^{2b}}+b^{(4a^2)^{2a}}\Big)^{\frac{1}{2}}\geq ...
5
votes
2answers
162 views

How many solutions to each of the equations $2^x+3^y=5^z$, $2^x+5^y=3^z$, $3^x+5^y=2^z?$

Let $x,y,z\in\Bbb{N}$ How many total solutions are there to each of the three distinct equations below? $$2^x+3^y=5^z \tag 1$$ $$2^x+5^y=3^z \tag 2$$ $$3^x+5^y=2^z \tag 3$$ I found 3 solutions to ...
5
votes
1answer
178 views

Is there a $k$ such that $2^n$ has $6$ as one of its digits for all $n\ge k$?

It is true that every power of $2$ of the form $2^{6+10x}$, $x\in\mathbb{N}$, has $6$ as one of its digits. Something more is true, the last two digits are either $64$ or $36$. The OP suggests that "....
5
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0answers
501 views

Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications?

There are the following common notations: Sums: $$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$ Products: $$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$ Is there a (theoretical) one for: Exponentiation (...
5
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1answer
98 views

Can this type of limit even be evaluated?

This is going to be a long question; please bear with me. We are familiar with the notations $\Sigma^{n}_{k=0} a_k$ and $\Pi^{n}_{k=0}a_k$ for the sum and product of the finite sequence $\{a_n\}$. I'...
4
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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,...
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*}\...
4
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0answers
340 views

Prove that $a^{4b}+b^{4a}\geq \frac{1}{2}$

Inspired by a problem of Vasile Cirtoaje I propose this : Let $a,b>0$ such that $a+b=1$ then we have : $$a^{4b}+b^{4a}\geq \frac{1}{2}$$ I compute the derivative of $f(x)=x^{4(1-x)}+(1-x)^{4x}...
4
votes
1answer
85 views

What's equal the below power nested radical?

it is well known that $$\frac{2}{\pi}=\sqrt{\frac12}{\sqrt{\frac12+\frac12\sqrt{\frac12}}{\sqrt{\frac12+\frac12\sqrt{\frac12+\frac12\sqrt{\frac12}}}{\sqrt{\frac12+\frac12\sqrt{\frac12\cdots}}}}}$$ ...
4
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0answers
84 views

How can you use mathematical notation of a logarithm to distinguish between the possible results?

What I've just asked might appear ambiguous and I suppose I could clarify myself with an example: e to the 0 is 1 as it should be, but e to the 2πi is 1 at the same time, so the base e logarithm of 1 ...
4
votes
2answers
102 views

complex number to a power divisible by 6

I actually have a follow-up question to this post -- given that n is a positive integer such that $z^n = (z+1)^n = 1$, I need to show that n is divisible by 6. I can now show that $z$ and $z+1$ both ...
4
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0answers
88 views

Find the $least$ number $N$ such that $N=a^{a+2b} = b^{b+2a}, a \neq b$.

When I graphed the relation $a^{a+2b}=b^{b+2a}$ , it gives a graph similar to $y=x$. However, the question explicitly states that $a \neq b$. So does that mean that no such $N$ exists ? What happens ...
4
votes
1answer
80 views

Is it true that $z^{w+1}=z^w\cdot z$ where $z,w\in \Bbb{C}$?

Is it true that $z^{w+1}=z^w\cdot z$ where $z,w\in \Bbb{C}$? The original question didn't state whether $w\in \Bbb{C}$, but I guess that would be the case, and it is a more inclusive approach. I ...
4
votes
1answer
146 views

Solving for the exponent of a power sum

Let $x$, $y$, $z$, $t$, be real positive numbers. Is it possible to solve $t$ from the following equation, if $x$, $y$, and $z$ are known? $$ x^t + y^t=z $$ If an exact solution is not possible, are ...
4
votes
1answer
71 views

Square Roots: Variables with Exponents.

Alright, so let me get this straight: $\sqrt{x^2} = |x|$ $\sqrt{x^3} = x\sqrt{x}$ $\sqrt{x^4} = x^2$ $\sqrt{x^6} = |x^3|$ Are these correct?
4
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0answers
74 views

Why can $x^0$ sometimes be simplified to 1 even when x can equal 0?

For example, the Taylor series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. It seems like it should be indeterminate or undefined at $x=0$, since the first term would contain $0^0$, but it's not ...
4
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0answers
110 views

How does one solve $y^y-x^x=x$ for $x$ as a function of $y$?

In order to find the answer to this question I started thinking that as a first step to obtain the first and second column, one would have to solve the equation: $$y^y-x^x=x$$ for $x$ as a function ...
4
votes
1answer
49 views

For even integers $a$ and $b$, since $(x^b)^{1/a}$ and $(x^{1/a})^b$ have different domains, does it follow that $x^{b/a}$ is not well-defined?

Let $f(x)=(x^b)^{1/a}$ and $g(x)=(x^{1/a})^b$, where $a,b$ are even integers. The domains are clearly different. $\operatorname{dom}\{ f\}=\mathbb{R}$, while $\operatorname{dom}\{ g\}=\mathbb{R}^+_0$. ...
4
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0answers
161 views

Iterations $n, n^n, (n^n)^{(n^n)},…$

(Note: I'm reposting this, as I posted the original too late in the evening to gain anyone's notice.) A contest problem (#2 on the 2010 Virginia Tech Math Competition) proffers the solver the ...
4
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0answers
119 views

Definition of $0^z$, $z$ complex

Usually we define the function $a^b$ over the complexes using the exponential function, as $e^{b\log a}$. This function has some issues with multivalued-ness, but it still more or less satisfies $a^ba^...
4
votes
1answer
72 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 which ...
4
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0answers
206 views

Convergence in Growth/Decay of Sum Odd and Sum Even Exponentiated Terms

where $n \geq 2 $ Given that a function containing an odd number of exponentiated terms as follows $$\left(\frac1{n}\right)^{\left(\frac1{n+1}\right)^{.^{.^{.^{\left(\frac1{n+2m+1}\right)}}}}} $$ ...
3
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0answers
43 views

Solve the inequality $ a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc$

Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$ a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc\quad (1)$$ I have a proof : I was thinking for an alternative proof considering by example Young's ...
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{...
3
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0answers
156 views

Conjectured upper bound $\frac{\Gamma(x+n)}{\Gamma(x)}$ and limit at infinity

Let $x\geq 1$ a real number and $n\geq 3$ a natural number then we have : $$\frac{\Gamma(x+n)}{\Gamma(x)}\leq \Bigg(\frac{nx+\Big(\frac{n(n-1)}{2}\Big)}{\frac{n}{2}}-\Bigg(x^x(x+1)^{x+1}(x+2)^{x+2}...
3
votes
0answers
61 views

Prove that $a^{2b}+b^{2a}\leq 1$ if $a,b>0$ such that $a+b=\frac{1}{2}$

let $a,b>0$ such that $a+b=\frac{1}{2}$ then we have : $$a^{2b}+b^{2a}\leq 1$$ My try : I have tried to apply Bernoulli's inequality : $$f(a)=a^{2(\frac{1}{2}-a)}=(1+a-1)^{2(\frac{1}{2}-...
3
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0answers
64 views

Proof about eigenspace of eigenvalue in power of matrix

Upon studying the Jordan normal form I came across the problem of determining the Jordan normal form of powers of a single base matrix and in that context I was wondering what happens to the ...
3
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0answers
103 views

Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and ℝ?

Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and ℝ? So far, I have only found this: http://andrusia.com/math/preliminaries/...
3
votes
0answers
249 views

How many pairs of adjacent numbers each have a prime number of factors?

Let $a,b\in\Bbb{N}\,|\,a+1=b$ and, $a$ and $b$ each have a prime number of factors Question: how many pairs (a,b) satisfy these conditions? I recommend trying this on your own first. There is a ...
3
votes
0answers
87 views

Axiomatizing the (positive-based) exponential function

The real-valued function $\langle x,y\rangle\mapsto x^y$ can be defined in multiple ways, and possible definitions vary somewhat, depending on the desired domain of definition. For instance, if one ...
3
votes
0answers
165 views

Can Pascal's Triangle be expressed as a exponential equation?

Pascal's triangle seems to follow a pattern. 11row-1 outputs PascalTriangle(row) while ...
3
votes
0answers
109 views

Inequality with exponent variable

Is there any way to isolate $n$ algebraically in this inequality? I've managed to find it graphically, but I'm lost on how I would apply algebra to this question. $$10000+800n>10000\times1.05^n$$
3
votes
1answer
35 views

Is there a name for an equation in which different variables are multiplied together? e.g. $xz+2x-5yx=21$

This is quite a simple question I suppose, but i'm trying to categorise equation types in my mind. From what I know, linear equations can have any number of variables in them, but in each term there ...
3
votes
1answer
199 views

Meaning of imaginary part of $\int_0^6 e^{x^3} dx$

My question is the title itself. How can it be possible that integral of real numbers can have an imaginary part?
3
votes
1answer
55 views

Number of integer triples to exponential equation

I'm taking a class on number theory and this is one of the problems my professor gave. How many ordered integer triples $(x,y,z)$ are there such that $x^y-y^x=2017\times z$, where $x,y$ are less than ...
3
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0answers
1k views

Comparing Large Exponents with different bases.

How to compare large exponents with different bases? Is there any way to roughly approximate their values? For example, sort the elements of list below based on their magnitude. $381600^{809197}, ...
3
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0answers
635 views

Mapping exponential functions in polar coordinates

I tried mapping power functions onto the polar plane (i.e. converting x,y into r and $\theta$). I was successful with power functions representing $y=ax^n$ by $$r=\sqrt[n-1]{\frac {\tan(\theta)}a}\sec(...
3
votes
0answers
476 views

Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it ...

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