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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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166 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 ...
7
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285 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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157 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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284 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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322 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{^} (...
4
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82 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
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81 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
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234 views

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

Let us define these two sequences as follows: $a_0=1$, $b_0=c$ $a_{n+1}=x^{a_n}$, $b_{n+1}=x^{b_n}$ $c\ne x^c$ $x,c\in\mathbb C$ Is it possible for $a_n$ to diverge but $b_n$ to converge under ...
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48 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 ...
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73 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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376 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 (...
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109 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
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139 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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116 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^...
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121 views

Is $x^{\frac 12}$ the same as sqrt(x)

maybe this question is very simple and clear and trivial to everbody. but right now i'm not sure. the equotation $$ x^\frac{1}{2} = \sqrt{x} $$ is only true whenever $ x \geq 0 $ right? the square ...
4
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203 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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81 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 ...
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124 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 ...
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62 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. $$10000+800n>10000\times1.05^n$$
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256 views

What is the middle digit of $9^{99}$?

Find the total number of digits and the digit in the middle of $9^{99}$, $\it{without}$ actually calculating any other digit of the number. PS: according to defuse.ca/big-number-calculator.htm, the ...
3
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911 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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459 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
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422 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 ...
3
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874 views

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
3
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224 views

Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia $$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k n^{m+...
2
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44 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 ...
2
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68 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/...
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99 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 ...
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47 views

Power profile of permutation matrices

$P$ denote a matrix from set of all $\{0,1\}^{n\times n}$ permutation matrices. We know there is a $k\in\mathbb N$ such that $P^k=I$ the identity. Given an integer $t\in\mathbb N$ how many $n\times n$...
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72 views

Fermat's Last Theorem And Modular Arithmetic

I was browsing Reddit yesterday when stumbled upon an interesting math problem: a user tried to implement a "smart contract" that was irredeemable under the validity of Fermat's Last Theorem. ...
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170 views

Logarithm and exponent of real quaternions

The logarithm of a general quaternion is defined as $$log(q) =\left (\left|q\right|, \frac{\mathbf{v}}{\left|\mathbf{v}\right|}cos^{-1}\left(\frac{r}{\left|q\right|}\right)\right),$$ in $(r,\mathbf{...
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39 views

Integral of a convination of nested exponentials.

I would like to calculate the following integral $$ \displaystyle\int_0^{T} e^{-s\left[t+\frac{1}{\mu}(e^{-\mu r }- e^{-\mu(r-t)})\right]} dt$$ where $s,\mu, r$ are constant values. What I have ...
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32 views

“Derivative-like” Operator such that $H(f(x)^{g(x)}) = H(f(x))^{H(g(x))}$

Given 2 functions $f(x),g(x)$ we have: $$D(f(x)+g(x)) = D(f(x))+D(g(x))$$ And $$D^*(f(x)g(x)) = D^*(f(x))D^*(g(x))$$ where $D^*$ is the multiplicative derivative: $$D^*(f(x)) = lim_{h\rightarrow 0} {\...
2
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0answers
126 views

Convergence or Divergence of Digital Exponentiation

The digital sum of a number is the sum of its digits, and it's similarly defined for digital products. Let's define the digital exponentiation of a number to be the evaluation of the exponentiation ...
2
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48 views

sum of $p^{\text{th}}$ powers of a set of $n$ variables $x_i$

Suppose we have n variables $x_1, x_2, ..., x_n$, and denote the $p^{th}$ power sum of these variables $S_p$ $S_p(x_1, ..., x_n) = \sum_{i=1}^n x_i^p$ Is there a method to compute (or approximate) ...
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44 views

Complex exponentiation in Wolfram Alpha

I don't well understand how WA performs the exponentials in the complex field. I see that a simple case as $(-1)^{2/3}$ is treated as $((-1)^{1/3})^2$ so it gives (for the principal values): $$ (-1)^{...
2
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0answers
159 views

Exponential of a symmetric, block tridiagonal matrix (with zero on the diagonal)

Is there an analytic result for the exponential of a symmetric, tridiagonal matrix (the diagonal can be zero, if this helps). Moreover, if it simplifies the result, the $\alpha$'s matrices can be (...
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0answers
100 views

The multivalued behaviour of complex exponential $z^\lambda$

On Gustav Doetch's Introduction to the Theory and Application of the Laplace Transform, it says: The power series $\sum_{n=0}^\infty a_nz^n$ converges on a circular disc. Replacing the integers $n$ ...
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140 views

How to define matrix power

I am currently writing a scriptum struggling with the definition of matrix power. Precisely, let $ \mathbb A \in \mathbb C^{n, n} $ and $ p \in \mathbb R $. I Currently have: If $ p \in \mathbb N $ ...
2
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0answers
127 views

Integral of the product of a power function and an arbitrary exponentiated function

I was only able to find integral tables that solve $$f(t)=\int t^c e^{kt}dt$$ but the integral I'm trying to solve has a function, not a constant, for the exponent: $$f(t)=\int t^c e^{g(t)}dt$$ Is ...
2
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0answers
158 views

Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$

I'm trying to show that for $b>1$, $x>0$ and $x$ irrational, that $$\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$$ I know this follows immediately if we define $b^x=\exp(x\log(...
2
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0answers
40 views

Exponentiation - I can do this?

Euler showed that: $$ e^{i\pi} + 1 = 0 $$ So I thought: $$ b \in \mathbb R,i = \sqrt{-1} \\ n^{bi} = e^{bi\ln n}\\ b = 2k\pi\\ e^{bi\ln n} = e^{2k\pi i\ln n} = (e^{i\pi})^{2k\ln n} = ((-1)^2)^{k\ln ...
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83 views

When is $0^0 = 1$ inconvenient? I heard sometimes $0^0 = 0$ may help.

Don't get me wrong: my question is NOT claiming $0^0 = 1$. I understand it's indeterminate. Many articles show that defining $0^0 = 1$ is (just) convenient and I completely agree. However, I've heard ...
2
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0answers
57 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / \...
2
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0answers
91 views

Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the $lim_{n\...
2
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184 views

How many zeroes would be there at the end of $11^{(5!)!}-1$?

$$11-1=10 \\ 121-1=120 \\ 1331-1=1330$$ Now it can be seen that the tens digit increases by 1 at each increment of exponent. So, only in case of $11^{10}$ the tens digit is zero and the units digit ...
2
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0answers
228 views

Find the value of a^b?

Given a series as f(n)=1^1 * 2^2 * 3^3 * ......n^n since n can be very large Find the value of f(n)/f(r)*f(n-r) and output it modulo m where m is any prime. Now My approach is f(n)=1^1%m * 2^...
2
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0answers
505 views

Commuting exponentials of non-commuting matrices

For two non-commuting matrices $A,B \in M(2,\mathbb{K})$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, can be shown that: $$ e^C=e^{A+B}=e^Ae^B=e^Be^A \iff \begin{cases} \mbox{det}A'=\mbox{det}A-\left(\...
2
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0answers
227 views

What does it mean to have an irrational/imaginary exponent and is there a way to calculate the latter?

In exponentiation, we are told that raising something to an integral power (n, say) means multiplying it with itself a total of n times, if n is non-negative. And we also learn fairly early on that ...
2
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0answers
431 views

If $f$ is holomorphic, then there is a holomorphic function $h$ such that $e^{h(z)}=f(z)$

Let $f:G\to\mathbb{C}$ denote a holomorphic function over a star-shaped domain $G$ and $f\ne 0$ on $G$. I want to show that it holds $\frac{f'}{f}$ is holomorphic There is a holomorphic function $h:G\...