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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31 views

Can decay percent be less than $-100\%$?

I have this problem: $$A=A_0\left(1+\frac{r}{n}\right)^{nt}$$ where $A=100$, $A_0=25$, $n=1$, and $t=2$. This leads to \begin{align*} (1+r)^2=4 &\quad\Rightarrow\quad |1+r|=2\\ &\quad\...
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48 views

Prove $\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$

I have seen this equation: $$\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$$ As you can see the numerator switched with the denominator and I wonder how. I know ...
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23 views

Why must the base and argument of a logarithm be positive? [duplicate]

A counter-example (based on my misunderstanding) is, say, if $(-3)^3=-27$ then why is it not the case that $log_{-3}(-27)=3$?
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Using fast modular exponentiation [closed]

I want to work out the last three digits of 1256^13 using fast modular exponentiation but I'm not clear on how to do this and am very confused about the process. Would really appreciate it if anyone ...
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2answers
52 views

Modular Arithmetic and repeated exponentiation

I was messing around with mod and repeated exponentiation and noticed that if we let $P_n(k)$ denote repeated exponentiation by $n$, $k$ times then, $$\text{mod} \ b : a^{P_n(k)} \equiv a^{P_n(k-1)} \...
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2answers
24 views

Doubt Regarding Notation with Exponents.

Consider $a^{b^c}$ without any brackets. My question is, if the brackets are not mentioned, what is the convention regarding the position of the brackets?
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1answer
30 views

Can a unit of currency be taken to n power? [duplicate]

I've been building a unit & rate library for a forex trading algorithm and I realized I didn't have an answer to this question: Can currencies be taken to the Nth power? Unlike physical units, ...
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27 views

When does the inequality hold? $\frac{(6x)^3(2x+1)^2}{2x(2x-1)} < \sqrt[2x]{(6x+1)^{6x+1}}, \quad x \in \mathbb{R}^+$

For the following inequality, I can see numerically that it will hold at around $x > 1.22$. \begin{align} \frac{(6x)^3(2x+1)^2}{2x(2x-1)} < \sqrt[2x]{(6x+1)^{6x+1}}, \quad x \in \mathbb{R}^+ \...
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51 views

Combinatorial proof that $2^{n+1}-1=1+2+4+\dots+2^n$ [duplicate]

Let $S_k$ be a set with $k$ elements. Then $S_k$ has $2^k$ subsets. So the formula $2^{n+1}-1 = \sum_{k=0}^n 2^k$ can be translated to $$\text{number of *nonempty* subsets of $S_{n+1}$} = \sum_{k=0}^n\...
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Scaling as a repeated shift

I was wondering if the following derivation of the scale operator starting from the shift operator is good. Everybody knows that an operator $T_a$ acting on a function $f$ as $$ T_af(x)=f(x+a) $$ can ...
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65 views

Prove that an inequality holds for $\alpha \ge 2$

It's related to my own question Refinement inequality of : $\sqrt{x}+x^{\frac{x}{x+1}}\geq x+1$ Let $x\geq 5$ be a real number then we have : $$\frac{x^2+1}{x+1}\Bigg(\frac{x^{\frac{x}{x+1}}}{x^{\...
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53 views

How to solve where an equation where the variable is in the exponent?

My question is how to solve this problem specifically, but this type of problem generally? $$ 2^{n/8} - n < 0 $$ EDIT I should have probably been more specific. This is a computer science ...
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56 views

Refinement inequality of : $\sqrt{x}+x^{\frac{x}{x+1}}\geq x+1$

Related to New bound for Am-Gm of 2 variables we have : Let $x\geq 5$ be a real number then we have : $$\sqrt{x}+x^{\frac{x}{x+1}}\geq \frac{x^2+1}{x+1}\Bigg(\frac{x^{\frac{x}{x+1}}}{x^{\...
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1answer
52 views

Meaning of “$\exp[ \cdot ]$” in mathematical equations [duplicate]

I am reading book "Fuzzy Logic With Engineering Applications, Wiley" written by Timothy J. Ross. I am reading chapter 7 and in this chapter, "Batch Least Squares Algoritm" has been defined. It ...
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1answer
57 views

Solve nonlinear equation

How to solve nonlinear equation: $$x+2.1*\frac{100}{1+e^{(10-q(x))/3}}-2=0,\\ here \quad q(x)=\frac{100}{1+e^{(10-x)/3}}$$ Are here any numerical method suitable to solve or any package? I tried ...
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3answers
46 views

Sum of digits of sum of digits of powers of 12345

Sum of digits of $12345$ is $1+2+3+4+5=15$. The sum of digits of sum of digits is $1+5=6$. I have plotted the sum of digits of powers of $12345$ with blue dots (x-axis is the power). As the average ...
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1answer
27 views

Is there a notation for summing a non-variable set of exponents

Given a static (not variable) range of integers, such as One Through Five, I have this equation. $b^1$ + $b^2$ + $b^3$ + $b^4$ + $b^5$ Is there a shorter way to write that, knowing that this is not ...
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1answer
49 views

Is it common to write $f'^{2}(x)$, particularly when $f$ is a trigonometric function?

It has been said many times that the notation for exponentiated or inverted trig functions, e.g.: $$\sin^2(x), \tan^{-1}(x), \csc^3(x)$$ is confusing, ugly, and terrible in general, but nevertheless ...
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53 views

Proving $(ab)^{1/n}=a^{1/n}b^{1/n}$.

In this book: Lang, Serge. "Basic Mathematics" (p. 71), appears this theorem and proof. Theorem 1. Let $a$, $b$ be positive real numbers. Then, $$(ab)^{1/n}=a^{1/n}b^{1/n}$$ Proof. Let $r=a^{1/n}$ ...
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How do I solve for $t$ in this question? $5^{2t+2}-{100}^{2t}=625$

How do I solve for $t$ in this question? $5^{2t+2}-{100}^{2t}=625$ I have tried to express the LHS in terms of 2 and 5 but I don't seem to find any useful result. Any help will be appreciated. ...
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39 views

Tetration fractal algorithm

In the code example at http://code.activestate.com/recipes/577917-tetration-fractal/ the author seems to implement the operation of a complex number, $z=x+iy$, raised to the power of itself, $z^z$, as ...
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51 views

Simplifying Ramanujan's limited infinite root

Background I was playing around with Ramanujan's infinite root : $$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}}$$ You can write any number as this sequence by limiting the number of square roots in ...
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166 views

On a symmetric equation over the integer lattice that involves the Euler's totient function

I would like to know hints or a proof, or counterexamples, for the conjecture that I've stated in the Question below. I'm interested in this in an attempt to continue the study of a question that I've ...
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1answer
29 views

Solving an equation with a variable as both a part of a term and exponent

The problem : $$ 60 + 10x=2^x$$ I've tried using Logarithms which leads to : $$\log(60+10x)=x\log 2$$ The $\log(60+x)$ is the tricky part
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Basic exponent problem from facebook

Given : $$\sqrt{\sqrt{(-4)^2}}$$ a) $2$ b) $-2$ c) $\pm2$ d) $2i$ e) $-2i$ f) $\pm 2i$ My opinion, the answer is $2$. Because the result of the root can't be negative. For example if we have $\...
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1answer
48 views

Prove $a\cdot 2^x+b\cdot 3^x+c\cdot 7^x$ has at most two real solutions for $a,b,c\ne 0$

Prove $a\cdot 2^x+b\cdot 3^x+c\cdot 7^x$ has at most two real solutions. I can prove a variant with two terms in the following way: $$a\cdot 2^x+b\cdot 3^x=0$$ $$-\frac{a}{b}=\left(\frac{3}{2}\right)^...
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146 views

Trying to _really_ understand exponents…

I am a programmer, but grew up with a pretty weak math education. While I can cobble together enough understanding to build something like these charts solo from raw data, my level of true math ...
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51 views

Why the first $x$ decimal places of $(5+\sqrt{26})^{x}$ are following a pattern?

$\sqrt{26}$ is irrational number, so the decimal places should show no pattern. But $(5+\sqrt{26})^{x}$ has these values: ...
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1answer
16 views

Formula for an exponent

Is there a formula for $n$ in an equation of the form: $$a^n+b^n=c$$ where $a,b,c \in \mathbb{R}$ and $n \in \mathbb{N}$? Is there any theorem that say something about this kind of a problem? ...
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24 views

exponent rule when dividing the same exponents

I ran into a bit of confusion when applying the exponent rule: $x^a/x^b = x^{a-b}$ Then when $4^x/2^x$ why does it equal $2^x$ if we apply the rule above then wouldn't it be: $(4/2)^{x-x}$ ? Thank ...
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15 views

Matrix equation with powers $A\cdot X^T=B$

Given the matrix equation $A\cdot X^T=B$, isolate the matrix $X$ and identify the order of the matrices $A$ and $B$ such that $X$ has order $3\times5$ I came across these matrix equations, and I ...
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17 views

Simplify irrational number in exponent [duplicate]

Is there a way to simplify an expression like $2^\sqrt{2}$, where the exponent is irrational, or is this already in simplest form? Also, would there be a way to estimate this value?
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34 views

Infinite tetration for complex domain [duplicate]

$$f(z)=z^{z^{z^{.^{.}}}}$$ $$for\space z \space \in C\rightarrow C$$ Can we define the range of this function(convergence-divergence specifically) without taking help from fractals? I checked out ...
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1answer
35 views

Combinatorial proof of $1+2(\sum_{i=0}^n 3^i)=3^{n+1}$

I have this workbook of proofs that I've been trying to finish for a couple of months now. There is this problem in it that requires me to prove $1+2(\sum_{i=0}^n 3^i)=3^{n+1}$ using combinatorial ...
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2answers
61 views

Counter-intuitive inequality : $a^{\frac{a^2}{a^2+b^2}}b^{\frac{b^2}{a^2+b^2}}+a^{\frac{a}{a+b}}b^{\frac{b}{a+b}}\leq \frac{2(a^2+b^2)}{a+b}$

I'm studing the following inequality : Let $a,b>0$ then we have :$$a^{\frac{a^2}{a^2+b^2}}b^{\frac{b^2}{a^2+b^2}}+a^{\frac{a}{a+b}}b^{\frac{b}{a+b}}\leq \frac{2(a^2+b^2)}{a+b}$$ It's related to ...
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77 views

Relation between $x^{x+1}$ and $(x+1)^{x}$, $x \in \mathbb{Z}$

So say that we have a pair $(x^{x+1},(x+1)^{x})$ for all $x \in \mathbb{Z}$. Is there any correlation between the members of this pair? Or are they not related?
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i to the power of i and other complex exponentials

After stumbling accross $i^i$, I have been become quite obsessed with complex numbers and especially complex exponentials. This even increased after realising that $i^i = e^{-\pi(2k + \frac{1}{2})} $...
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0answers
42 views

'Multiplicative' integration and 'Riemann products'

Warning! This question has little rigour and is entirely hand wavy crazy blue-sky speculative thinking so I apologise in advance. I was thinking about the gamma function and how it interpolates the ...
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15 views

Quaternion exponent and logarithm to non-standard base

Given that the exponent of a quaternion $q = (w, \vec{v})$ with base $e$ is defined as \begin{align} e^q = e^w (\cos |\vec{v}| + \frac{\vec{v}}{|\vec{v}|} \sin |\vec{v}|) \end{align} and the ...
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1answer
60 views

How to find e^x using Newton Raphson method??

Is there a way to find e^x using Newton Raphson method, I found this link: Iterative refinement algorithm for computing exp(x) with arbitrary precision but I don't know how to implement it. Please ...
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35 views

What is the fastest way to compute the exponential (e^x) of a number? [duplicate]

I'm making e^x function in C language by using the Taylor series but it is very slow. It takes a lot of time to compute the value of e^x. Is there any faster way to compute e^x?
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59 views

How to prove why $(2n + 1)^{16} \bmod 64 = 1$?

The title says it all :-) I verified experimentally this and I'm stuck with the 16th power of $(2n + 1)$: $$ 65536n^{16} + 524288n^{15} + 1966080n^{14} + 4587520n^{13} + 7454720n^{12} + 8945664n^{11}...
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How to find the nth term and limit of sequence in the form of $a_{n+1}=b\ln{(a_n^2+c)}+d$

My problem is $a_{n+1}=\frac{2011}{3}\ln{(a_n^2+2011^2)}-2011^2$ with given $a_1=a$. I have no idea at all about what I should do. PLEASE HELP
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84 views

For $ a>b>1$, prove that $a^{b^a}>b^{a^b}$

Given that $a>b>1$, show that $a^{(b^a)}>b^{(a^b)}$. I've proved that the case $b\geq e$ holds as follows: $$\ln\ln a^{(b^a)}-\ln\ln b^{(a^b)}=a\ln b+\ln\ln a-b\ln a-\ln\ln b$$ Let $t=a-b&...
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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\...
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1answer
50 views

Why it were conjectured that $e^{e^{^e{^{79}}}}$ is not an integer only for $n=79$ ? any non trivial characterization?

I'm confused that why exactly and what is the reason to conjecture that $e^{e^{^e{^{79}}}}$ is not integer , why not for example with $n=87$ or any other prime $p$ ?Is this number special ? or is ...
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1answer
53 views

Can $\lim_{h\to 0} (1+h)^{[\frac{x}{h}]}$ be an equivalent definition of the exponential function?

Here, take $[x]$ to be the smallest integer function. I like this more than $\lim_{n\to \infty} (1+\frac{x}{n})^n$ because it seems to make the property $\exp(x+y)=\exp(x)\exp(y)$ obvious. This is ...
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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}-...
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1answer
19 views

finding partial derivative of two equal functions gives different results

$e^{\ln[x+w]} = xw$ I hope we agree left side is equal to right side. Finding partial derivative $w$ given right side is simply $x$ But when I used online partial derivative calculator and I tried ...
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3answers
44 views

Interpretation of multi-exponent x^y^z ($x^{y^z}$ or ${x^y}^z$)

Is there any generally accepted or written rule specifying the interpretation of a multi-exponent expression written in this simple style (by user, no latex): x^y^z?...

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