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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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3answers
91 views

Is $i = e^{\frac{\pi}{2}e^{\frac{\pi}{2}^{.^{.^.}}}}$? [on hold]

I came up with this and I am wondering if it is true, because it seems illogical that $i$ can be made from an infinite power tower of reals. The way I found this is the following: $$i=e^{\frac{\pi}{2}...
3
votes
2answers
73 views

Explicit formulas for $\operatorname{Re}(z^n)$ and $\operatorname{Im}(z^n)$

I'm looking for a closed formula for the real and imaginary part of $z^n = (u + iv)^n$. We have $$\operatorname{Re}(z^{n+1}) = u\operatorname{Re}(z^{n}) - v\operatorname{Im}(z^{n})$$ $$\...
1
vote
0answers
19 views

Maximum of exponential tower

First I introduce a notation similar to $\sum_{i=1}^n a_i$ for exponentiation. I.e. for any (potentially infinite) sequence $a_i$ we define $$ ES_{i=1}^n a_i = \left\{\begin{matrix} a_1 &...
1
vote
1answer
35 views

How does this method work to find prime numbers?

I'm curious about this pattern that I saw while adding many powers of two together, and then taking the prime factorisation of each result, and I'm curious as to why this occurs. Here is the pattern: ...
0
votes
1answer
56 views

Simplify $x^n + x^{n-1} + … + x^1 + 1$ [duplicate]

Can I somehow simplify $x^{n-1}+x^{n-2} + ... + x^{2} + x^{1} + 1$? I would like to have an explicit formula for that sum, but could not figure out a way to do so. Could you help me? Thanks!
4
votes
1answer
85 views

$\int_{-\infty}^{\infty}{x\choose t}dt=2^x$ for $x\geq0$??

I noticed that on Desmos, for $m>0$ and $x\geq0$, $$\int_{-m}^{m}{x\choose t}dt$$ closer and closer approximated $2^x$. So, does $$\int_{-\infty}^{\infty}{x\choose t}dt=2^x$$ Assuming that ${x\...
0
votes
1answer
14 views

Rearranging Poisson distribution variables: lambda and exponents

Another stackoverflow question provides the following logic to get from an expected value for Xg(X) where random variable has a Poisson distribution: \begin{align} E[X g(X)] &= \sum_{x=0}^\infty ...
2
votes
3answers
77 views

Factor $10^6-1$ completely

I know kind of a very elementary method to factor this number. Consider the following: $$10^6-1 = (10^3-1)(10^3+1)=9 \times 11 \times (10^2+10+1)(10^2-10+1) = 9 \times 11 \times 111\times 91$$ I would ...
1
vote
1answer
25 views

Exponential growth with growth factor <2 [duplicate]

Is there an $a$ with $1<a<2$ such that $y=a^x$ can be bounded (or eventually dominated) by a polynomial?
0
votes
1answer
45 views

Is it possible to reduce an algebraic function to $1$s and $0$s?

I want to know where in this process I'm going wrong. Perhaps it's not even a valid thing to do...? Take a well-behaved function such as $f(x)=x \sin 2x$. I want to turn this into a new function $g$ ...
0
votes
1answer
44 views

Simplifying $\left\{\left[(2/9)^4\times(3/14)^4\right]^4:(-1/7)^2\right\}\times\left[(-5/6)^3:(5/18)^3\right]^3$ [closed]

$$\left\{\left[\left(\frac29\right)^4\times\left(\frac3{14}\right)^4\right]^4:\left[\left(-\frac17\right)^2\right]\right\}\times\left[\left(-\frac56\right)^3:\left(\frac5{18}\right)^3\right]^3$$ I've ...
31
votes
4answers
2k views

What is the reasoning behind this exponents question?

What is $3^{3^{3}}?$ Plugging $3^{3^{3}} $into the calculator gives 7625597484987. I believe because this implies that $3^{3^{3}}=3^{27}$, is this true? And plugging $(3^{3})^{3}$ gives 19683, ...
1
vote
1answer
40 views

Why is the algorithm for modular exponentiation by squaring considered as poly time?

As the link on Wikipedia says, and ive read it in many other books aswell, if we use squaring in exponentiation for Modular Exponentiation the complexity reduces and cuts down to O(log n), where n is ...
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votes
1answer
34 views

Help with simplifying an exponential equation

$$\frac{x^3y^{-1}-y^3x^{-1}}{x^{-1}y+xy^{-1}}\cdot\frac{x^{-2}+y^{-2}}{xy^{-3}-x^{-3}y}$$ I need to simplify the above. Can anyone provide me with a step-by-step solution to this problem? I tried ...
2
votes
2answers
49 views

How to represent $x^y$ = [Integer] + [Remainder]?

For example: $5^{\frac{1}{2}} = 2.23606\ldots = 2 + 0.23606\ldots$ Can we do this for $x^y$ in general? Motivation: a way of expressing the floor, $ \lfloor x^y\rfloor $, of an exponential $x^y$ ...
3
votes
1answer
72 views

Collatz “factorization”

The collatz conjecture states that every number eventually reaches $1$ under the repeated iteration of $$ f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \...
0
votes
1answer
30 views

Is there any number that satisfies $\lvert n\rvert > 2, n\in\Bbb{C}$ and $a^n+b^n=c^n$ where $a,b,c \in \Bbb{C}$

Fermat's Last Theorem has been proven, and this means that there are no integer values of $n$ that satisfy the equation $a^n+b^n=c^n$ where $a,b,c \in \Bbb{C}$ for $n>2$. But are there any rational ...
1
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2answers
107 views

Proof of : $x^{n}\equiv x^{\varphi(m)+[n \bmod \varphi(m)]} \mod m$

There is a lesser known generalization of Euler's Theorem. Here 'x' and 'm' are NOT COPRIME. I stumbled upon this on this site : http://cp-algorithms.com/algebra/phi-function.html The "derivation" ...
1
vote
2answers
36 views

Representation of any positive integer as a series $2^{n_1}-2^{n_2}+2^{n_3}-2^{n_4}+\cdots\,$?

Essentially the title: a friend stumbled upon this the other day, and we can't find anything on the internet about it. The $n_k$s are all distinct positive integers. As an example: $$45 = 2^6-2^5+2^...
1
vote
2answers
31 views

What properties of exponents and logarithms is used here?

In a solution to one of my assignment is given that: $$4^{log_{16} n} = n^{log_{16} 4} = n^{\frac12}$$ I understand the 2nd part, that's simple, but how was the first part achieved?
0
votes
0answers
19 views

Approximating exponential of differential operator acting on the exponential of a function

I am looking for an approximation of the following expression $$z(x)=\frac{\mathbf{e}^{-\nabla^2} \mathbf{e}^{f(x)}}{\mathbf{e}^{f(x)}}.$$ It is known than the exponential operator can be expanded ...
2
votes
2answers
68 views

Explicit Formula of Exponential of Companion Matrix

Let $A=\begin{bmatrix} a_k & a_{k-1} & a_{k-2} & ... & a_2 & a_1 \\ 1 & 0 & 0 & ... & 0 & 0 \\ 0 & 1 & 0 & ... & 0 & 0 \\ ... & ... &...
1
vote
1answer
30 views

complex exponentiation: evaluating $i^{1+i}$

I am trying to find evaluate the expression $i^{1+i}$. I know that, \begin{align} i^{i+1}&=\exp((1+i)\log(i)) \ \ \ \ \text{(where $\log$ is multivalued)} \\ &=\exp((1+i)(\ln|i|+i\arg(i)+2k\...
0
votes
2answers
18 views

How to show that they are equal?

Suppose A = $1/2^{100logn}$, and B = $e^{-log2*100*logn}$ I'm required to prove that A and B are equal, how should I prove this ? I tried applying some rules of logarithms that I have learned but I'm ...
3
votes
4answers
79 views

What is the correct solution of $x^{2/2}$?

Ok I get that $$\left(\sqrt x\right)^2 = x$$ and $$\sqrt{x ^2} = |x|.$$ I can explain this to myself when I insert $-1$ for $x$. With complex number $i$ I obtain those solutions. Fine. Now when we ...
0
votes
1answer
36 views

From $\forall p \in \mathbb{R} \lim_{x \to 1} x^p = 1$ conclude that $\forall p \in \mathbb{R} \, f(x) := x^p$ is continuous on $(0, + \infty)$

I am trying to solve exercise 9.4.4 from Tao's Analysis I. It says: Prove the following theorem: Let $p$ be a real number. Then the function $f : (0, \infty) \to \mathbb{R}$ defined by $f(x) := x^p$ ...
0
votes
0answers
12 views

Weighted and exponential distribution

I have no advanced mathematics knowledge and I would like to come up with a formula that would express my thinking. I have a pot of money of $1,000 that I would like to distribute to 3 persons ...
2
votes
4answers
62 views

Does the fact that $x^2=(x-1)(x+1)+1$ have a name?

Just curious about this pattern $$x^2 = (x-1)(x+1) +1$$ So: $$\begin{align} 1^2 &= \phantom{1}0\cdot\phantom{1}2+1 = 1 \\ 2^2 &= \phantom{1}1\cdot\phantom{1}3+1 = 4 \\ 3^2 &= \...
0
votes
1answer
36 views

Looking for formula or closed form for this series

I don't know if I should call this a series, or sum, or what. I am not very good at math, which is why I am here. But i can describe it. It is for a CS algorithm. I have a graph, with N nodes. N ...
0
votes
1answer
102 views

Subtracting exponents with same base

If $2^4 - 2^3 = 2^3$ and $2^5 - 2^4 = 2^4$, then is below a rule of subtracting exponents with similar base and exponents which are just $1$ away from each other? $$A^e - A^{e-1} = A^{e-1}$$ I will ...
1
vote
1answer
27 views

Imaginary number/exponent rules misconception

I must be doing something wrong in the following question and would appreciate clarification: Given that $i^2$= -1, and that k is a positive integer, what is the value of $i^{4k+2}$? My answer: $...
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votes
2answers
36 views

math problem with exponent [closed]

I have this formula for a game where you gain so-called "skills" depending on how many hits you make on a monster in the game. To go from skill 97 to 98, I can calculate the amount of hits by doing: ...
0
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0answers
30 views

Using Babylonian Numeration, Mayan Numeration, Base Conversions

I'm struggling with understanding and using the Babylonian system to solve problems as well as the Mayan system, and I struggle most with converting from one base to the next. For my course at the ...
0
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1answer
23 views

Algebraic Exponent GCSE Edexcel Exam Question. Help Please…

x = $2^p$ , y = $2^q$ (a) Express in terms of x and/or y, (i) $2^{p + q}$ (ii) $2^{2q}$ (iii) $2^{p -1}$ My Working Out: (i) x + $√y$ because $2^p$ is equal to x and I think $√2^q$ is equal to ...
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votes
1answer
62 views

Solving $\sqrt{x}^{\sqrt{x}^{\sqrt{x}}} = 2^{512}$. [closed]

I can't solve this question I have tried but I can't find any other websites that help. Thanks $$\sqrt{x}^{\sqrt{x}^{\sqrt{x}}} = 2^{512}$$ This is not infinite exponent just 2 times
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votes
3answers
86 views

How to find $\displaystyle\lim_{n\to\infty}\frac{2^{2^n}}{n!}$?

What I have done is $\frac{2^{2^n}}{n!}=\frac{2^{2^n}}{1 * 2 * \dots * n} \leq \frac{2^{2^n}}{n}$. Then I am stuck here. I wanted to do something like in this answer using Squeeze theorem. Is it right ...
3
votes
1answer
46 views

Why not define exponents as $x*|x|*|x|$…

Why are exponents defined such that $x^2 = x*x$, rather than $x^2 = x*|x|$? Doesn't this alternative definition simplify a lot of things? Currently, if you take $x^2=4$, $x$ can equal $2$ or $-2$. But ...
0
votes
1answer
38 views

Solve y(a^y-1)=b

Trying to solve this equation for $y$ leads me into horrible thickets of logarithms that also seem unsolvable. $$y(a^y-1)=b$$ $a$ and $b$ are constants. Is there a simple solution of the form: $$y ...
8
votes
1answer
79 views

Solve: $\log_3(5(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})+2^{64})=?$

$$\log_3(5(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})+2^{64})=?$$ At a first glance, it seems that I need to do this: $$\log_3((2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{...
2
votes
2answers
48 views

$\forall x \in \mathbb{R}^{+}, n \in \mathbb{N}$, prove that $\exists y \in \mathbb{R}^{+}$ such that $y^n = x$.

Q: Let $n$ be a natural number and $x$ a positive real number. Prove that there is a positive real number $y$ such that $y^n = x$. Is $y$ unique? \begin{align*} y^n &= x \\ y &= x^{1/n} \\...
2
votes
0answers
31 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 ...
0
votes
0answers
55 views

Finding the smallest prime factor of $\sum_{a=1}^N a^{k}$

It is linked to my previous question, but I wanted a ++ clear explanation: Suppose we have a huge number of that type with a huge $k$. $\sum_{a=1}^N a^{k} =1^{k}+2^{k}+3^{k}+...+N^{k}$ And we want ...
0
votes
0answers
48 views

Solve modulo of a power of a power

I saw many examples when you have a power of a power modulo something. Something like $7^{7^{7^{...}}}$ mod n And then you start changing the exponent to a modulo, like: $7^{7^{7^{...}} mod (m)}$ ...
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votes
1answer
51 views

combinatorial argument for Exponential Generating Function [closed]

Could you tell me how can I use a combinatorial argument to prove that $(e^x)^n = e^{nx}?$
3
votes
1answer
64 views

Why $(i-1)^{2i}\neq[(i-1)^2]^i$? [closed]

With real numbers, we have the Power of Power Law:$$a^{xy}=(a^x)^y.$$ However, this doesn't always work with complex numbers. For example, Mathworld states that: $$(i-1)^{2i}\neq[(i-1)^2]^i.$$ I ...
1
vote
1answer
43 views

Converting $i$ to exponential form

How would I convert the irrational number $'i'$ to exponential form, or $e^{i\theta}$? I'm working a little in De Moivre's theorem.
2
votes
2answers
73 views

Find when $2^{3^{4^{…^{n}}}} \equiv 1$ (mod $n+1$) is true

It is linked to my previous question, I haven't been given any clue for how to verify this modular equation: $2^{3^{4^{...^{n}}}} \equiv 1$ (mod $n+1$) How can I find the condition for $n$?
3
votes
1answer
57 views

Minimum of repeated iteration of $n^{-x}$

Consider the repeated iteration of the function $f(x) = n^{-x}$, (meaning the result from the first calculation is the argument for the next and so on) The first value of $x$ can be any positive ...
0
votes
0answers
93 views

$2^{3^{4^{…^{n}}}} \equiv 1$ (mod $n+1$)

I remember when I started learning modular arithmetics I found a tetration equation stated as follows $2^{3^{4^{...^{n}}}} \equiv 1$ (mod $n+1$) I am wondering how could this be proved, I tried this ...
1
vote
2answers
66 views

Solve power with negative exponent $\frac{125^{6}\times 25^{-3}}{(5^{2})^{-3}\times25^{7}}$

I am currently studying about exponents and powers for college calculus discipline. In the meantime I came across negative exponents, like this $25^{-3}$ and $(5^{2})^{-3}$. I have this calculation ...