Questions about exponentiation

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0
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2answers
204 views

Simplifying exponents, multiplication, and addition

How can you get $10^{n+1}$ from $9\cdot 10^n+10^n$? This is part of a proof I am working on.
0
votes
1answer
90 views

Negative even exponents of negative numbers

How can I calculate and prove this equation with mathematical terms: pow((-2), -2)=? I know that pow(1, -1) is equal to 1/1 by the way. Any idea, please?
3
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2answers
2k views

Growth of exponential functions vs. Polynomial

Will $2^x$ take over $x^{1000}$ ? I thought that exponential functions had the fastest growth rate, however, graphing it on wolfram alpha made it seem as if the initial behaviors of the two functions ...
2
votes
2answers
323 views

Is there a reasonable generalization of the falling factorial for real exponents?

The falling factorial is defined as: $$x^\underline n = \prod_{k=0}^{n-1}(x-k),\quad n\in\mathbb N$$ It can be used to define the binomial: $$\binom n k = \frac{n^\underline k}{k!}$$ And it ...
2
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3answers
177 views

Mathematical function for the powers

I have this formula $$\underbrace{2^{2^{2^{.^{.^{.^{2^2}}}}}}}_n$$i.e. where the total number of 2's is $n$. Is there any way to write it as a single mathematical function?
2
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1answer
147 views

Has a matrix block diagonal structure if and only if its exponential has it as well?

Obviously if $\mathbf{A}=\begin{bmatrix}\mathbf{C} & \mathbf{0} \\ \mathbf{0} & \mathbf{D}\end{bmatrix}$ then $e^{A}=\begin{bmatrix}\mathbf{e^C} & \mathbf{0} \\ \mathbf{0} & ...
1
vote
1answer
107 views

The power of a power

My teacher gets the following: $(x^{2})^{12-k} = x^{2k-24}$ Where I get the following: $(x^{2})^{12-k} = x^{24-2k}$ I'd like to think of $2(12-k)$ as $2*12 - 2*k$ or $-2k + 24$. Why/how am I wrong? ...
0
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2answers
669 views

Notation of inverse trigonometric functions and exponentiation [duplicate]

Possible Duplicate: $\arcsin$ written as $\sin^{-1}(x)$ I have worked a bit on trigonometry today, and something strikes me as inconsistent. In the book, the notation for the inverse sine ...
21
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1answer
1k views

Infinite tetration, convergence radius

I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite ...
1
vote
2answers
74 views

How to show $\sum_{k \geq 2} \frac{(-x)^k}{k!} \geq 0$ for large x

It's probably a very silly question. I could only show though that $$ \sum_{k \geq 2} \frac{(-x)^{k}}{k!} = \sum_{k \geq 0} \frac{(-x)^{k}}{k!} -1 +x= e^{-x}-1+x $$ which tends to infinity as $x ...
1
vote
4answers
761 views

How does $\exp(x+y) = \exp(x)\exp(y)$ imply $\exp(x) = [\exp(1)]^x$?

In Calculus by Spivak (1994), the author states in Chapter 18 p. 341 that $$\exp(x+y) = \exp(x)\exp(y)$$ implies $$\exp(x) = [\exp(1)]^x$$ He refers to the discussion in the beginning of the ...
1
vote
2answers
999 views

11th grade level exponential growth problem?

A certain strain of bacteria that is growing on your kitchen counter doubles every 5 minutes. Assuming you start with only one bacterium, how many bacteria could be present at the end of 96 ...
3
votes
2answers
216 views

How to find complex numbers $z,\lambda,\mu$ such that $(z^\lambda)^\mu\neq z^{\lambda\mu}$

Let $z$, $\lambda$, $\mu$ be complex numbers. Find a case where $(z^\lambda)^\mu$ is not equal to $z^{\lambda\mu}$. In our book, $a^b = \exp( b \cdot \operatorname{Log}(a) )$. ...
0
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4answers
1k views

Find out monthly growth based on yearly growth

Let's say I have a number, 10, and over a period of a year that number has increased by 100% to ...
3
votes
2answers
998 views

The inequality $b^n - a^n < (b - a)nb^{n-1}$

I'm trying to figure out why $b^n - a^n < (b - a)nb^{n-1}$. Using just algebra, we can calculate $ (b - a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) $ $ = (b^n + b^{n-1}a + \ldots + ...
4
votes
1answer
66 views

Find the absolute maxmium of the function $f(x) = x \cdot {e}^{-x}$

How can I find the absolute maximum of this exponential function? $f(x) = x \cdot {e}^{-x}$ I know that the first step is to take the derivative of the function, like so: ${f}^{\prime}(x) ...
5
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3answers
1k views

Examples of continuous growth rates greater than exponential

I read on Wikipedia that growth rate of a function can sometimes be greater than exponential. Can you give me some examples of such functions (preferably continuous ones)? Obviously $x^x$ grows ...
4
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3answers
150 views

Prove $\frac{(5^{x-1}+5^{x+1})^2}{25^{x-1}+25^{x+1}}=\frac{338}{313}$

Q. Prove $$\frac{(5^{x-1}+5^{x+1})^2}{25^{x-1}+25^{x+1}}=\frac{338}{313}$$ My try: expand and got: $$\frac{5^{2x-2}+2(5^{x^2-1})+5^{2x+2}}{5^{2x-2}+5^{2x+2}}$$ Now what? I find my pre-calculus ...
7
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5answers
1k views

How do I solve this analytically $3^x=9x$

One of my friends ask me how to solve this equation analytically $3^x=9x$. Looking at it I guess 3 is the answer and I also plot a graph of line $9x$ and the curve $3^x$, they intersect at 3. But, ...
66
votes
4answers
2k views

Complexity class of comparison of power towers

Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < ...
5
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2answers
157 views

How many positive roots does the equation $a^x=x^a$ have?

Let $a\in (1,e)\cup(e,\infty).$ I'd like to show that the equation $a^x=x^a$ has exactly two positive solutions, and one is larger and one smaller than $e.$ Is it even possible to show? I think I've ...
2
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2answers
75 views

Simplifying this exponential equation

I am wondering how does $$\frac{{{e^{zk}}}} {{{z^2} + 1}} = \frac{1} {{2i}}\left( {\frac{{{e^{zk}}}} {{z - i}} - \frac{{{e^{zk}}}} {{z + i}}} \right)?$$ I can see that $z^2 + 1 = (z + i)(z − ...
1
vote
1answer
180 views

Evaluating $0^0$ and its limit [duplicate]

Possible Duplicate: Zero to zero power From what I understand $0^0$ is indeterminate, yet when you evaluate $\lim\limits_{x\to 0}x^0$ you get 1 (given on wolframalpha.com). Something ...
4
votes
2answers
182 views

How exactly do exponentials work?

Hi all I know that $5^1 = 5$, $5^2 = 25$, $5^3 = 125$. But why is $5^{1.5} = 11.180339887498949$ ? How did we get the number $11.180339887498949$ ?
12
votes
6answers
1k views

Proving the inequality $e^{-2x}\leq 1-x$

How do I prove the inequality $e^{-2x}\leq1-x$ for $0\leq x\leq1/2$?
1
vote
1answer
150 views

Computing the power of real algebraic numbers

I'm looking for an efficient algorithm to compute the $n$-th power $\alpha^n$ of a real algebraic number $\alpha$ given by an interval representation for $n \in \mathbb{N}$. An interval representation ...
1
vote
2answers
210 views

Domains closed under exponentiation

Apart from $\mathbb{N}$ and $\mathbb{C}$, which other domains satisfy $\forall x, y \in D, x^y \in D$ ,i.e. are closed under exponentiation?
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2answers
318 views

Fixed points of $e^z$

How would one find the fixed points of $e^z$, where $z$ is complex (if there are any)? I feel this problem probably has a really obvious answer, and for some reason, I'm just not getting it. Thanks.
2
votes
3answers
105 views

Derivative of exponential function $\frac{d}{dx}a^x$

I am trying to compute simple derivatives of simple functions, but I got stuck on $\frac{d}{dx}a^x=(\ln{a})a^x$. I suppose the proof is a simple corollary of $\frac{d}{dx}e^x=e^x$, but I am unable to ...
0
votes
2answers
157 views

How to define $(-1)^{\frac24}$? [duplicate]

Possible Duplicate: Which step in this process allows me to erroneously conclude that $i = 1$ According to the definition of exponentials, $\displaystyle(-1)^{\frac{2}{4}}$ is equivalent to ...
2
votes
2answers
111 views

Using exponents when working with matrices

I was working on the following problem when I stumbled upon an oddity. If $X=P^{-1}AP$ and $A^3=I$, prove that $X^3=I$ My first approach was to cube both side which led to the following: ...
0
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3answers
366 views

$\lim_{n \to \infty}(1+\ln(1-\frac{1}{2n}))^n$

How would you find this limit ? $\lim_{n \to \infty}(1+\ln(1-\frac{1}{2n}))^n$ Thank you.
2
votes
2answers
180 views

Solving $x^\frac{1}{x}=y$ for $x$

I've tried everything: Taking $\ln$ of both sides, raising to the power $x$, nothing seems to work. Is there a way to solve this or am I going to have to use numerical methods instead?
0
votes
1answer
238 views

Recursive algorithm for calculating powers

I am working on a maths exercise and got this question: Make a recursive algorithm on the calculation of $x^p$, where $x$ is a real number and $p$ is a natural number of $n$ bits. I really don't ...
-2
votes
1answer
279 views

Smallest positive integer for equation

I am having trouble identify the smallest positive integer $n$ such that $(\frac{1+i}{1-i})^n = 1$ Can someone please throw on approach? (Also, please correct the equation in the form of Tex/Latex ...
7
votes
3answers
676 views

What we can say about $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$?

Problem: How we can strictly prove $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ can't be 2? Can $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ have the value expressed by complex numbers? (See below, in ...
8
votes
3answers
577 views

Is $n^n$ a perfect square or not?

If $n$ is an integer, how do you know whether $n^n$ is a perfect square, without a calculator? The actual question is: "how many integers between $1$ and $100$ inclusive, raised to their own power, ...
4
votes
2answers
274 views

Which is the “fastest” paper-pencil method to compare $\sqrt[17]{6}$ and $\sqrt[16]{4}$?

Which is the "fastest" paper-pencil method to compare (find which one is greater) $\sqrt[17]{6}$ and $\sqrt[16]{4}$? My analysis bought this whole thing down to comparing which is greater ...
0
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1answer
139 views

finding the decay constant

Given the following function, how does one rewrite the exponential part of the equation into $e^{-L/L_{0}}$, where $L_{0}$ is the decay constant ...
5
votes
2answers
214 views

Slope definition of $e$

A common definition of $e$ is given as $$e = \lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n}$$ which can be proven to be equivalent to $$e=\lim_{h\rightarrow 0}\ ...
51
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4answers
3k views

Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?

Problem: Find $x$ in $$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$ Trick: $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so, $x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and, ...
3
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2answers
295 views

Algorithm for computing powers

I was challenged by one of my fellow students to write a mini-library in the programming language called C that enables you to work with very large numbers (the numbers that the language offers ...
3
votes
2answers
114 views

limit when searching $a^x$

Sorry, my mathematical english vocabulary is not as vivid as I would like it to be, therefore my topic touches the main problem in searching limit for this function. $$\lim_{x \to 0} {\left ...
1
vote
0answers
83 views

Integer exponentiation algorithm for the special case $3^n$

Is there any known integer exponentiation algorithm to compute $x^y$ for the special case $x = 3$ which is faster than the general case algorithm found in [1], section 4.6.3? [1] D. E. Knuth, The Art ...
1
vote
1answer
189 views

Are numbers : $(-1)^{i} , 1^{-i} , 1^{i} $ transcendental numbers? [duplicate]

Possible Duplicate: What is the value of 1^i? According to Euler's formula : $e^{ix}=\cos x + i\cdot \sin x$ we may write : $$e^{i\cdot \frac{\pi}{2}}=i \Rightarrow \left(e^{i\cdot ...
45
votes
4answers
2k views

A new imaginary number? $x^c = -x$

Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
1
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1answer
273 views

How to solve for the equation $ax \exp(bx)=c$?

How to solve for the equation $ax \exp(bx)=c$? It is known that $x\geq 0$.
2
votes
1answer
211 views

Justifying a pair of inequalities involving the exponential function

I'm reading Fan Chung's Spectral Graph Theory. There's a pair of inequalities I don't know how to justify. Chung doesn't attempt to explain them, so maybe they're very obvious. Example 1.19 on page ...
1
vote
2answers
87 views

Logarithmic equations and their related exponential equations

I am learning about logarithms and I'd like some examples of the following: Examples where the logarithmic value is a different positive integer Examples where the logarithmic value is a different ...
10
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4answers
535 views

Which step in this process allows me to erroneously conclude that $i = 1$

I was playing around with imaginary numbers and exponents and came up with this: $$ i = \sqrt{-1} $$ $$ \sqrt{-1} = (-1)^{1/2} $$ $$ (-1)^{1/2} = (-1)^{2/4} $$ $$ (-1)^{2/4} = ((-1)^{2})^{1/4} ...