Questions about exponentiation
2
votes
0answers
28 views
Exponent p-value generated in Excel
Excel gave me a p-value of 1.44909E-09
Notice is does not say .09 but 09
This is confusing me, I am trying to analyze my data but am stuck at this point.
If it were E-9 it could be ...
0
votes
1answer
52 views
Why $\frac{1}{2i}(e^{i\omega t} - e^{-i\omega t}) = \frac{i}{2} (e^{-i \omega t} - e^{i\omega t})$
Let $i := \sqrt{-1}$, $f$ be the frequency ($\frac1p$), and $\omega := 2 \pi f$.
From page 3 here, why does $\frac{1}{2i}(e^{i\omega t} - e^{-i\omega t}) = \frac{i}{2} (e^{-i \omega t} - e^{i\omega ...
1
vote
1answer
33 views
Addition of numbers with same base but different exponents
The problem itself is $(288\sqrt{3})^{1/5}+(288\sqrt{3})^{4/5}$. The answer is supposed to be of the form $a+b\sqrt{c}$, but I have no idea how to simplify it.
12
votes
1answer
365 views
Can you raise $\pi$ to a real power to make it rational?
We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this:
Take $(\sqrt{2})^{\sqrt{2}}$
If it's rational, ...
1
vote
0answers
51 views
Generalisation of Lambert W function?
I want to solve an equation of the form:
$\exp(C / x) - 1 = D / (x + a)$
This seems to be almost in a form where I can express solutions in terms of the Lambert W function but I can't seem to figure ...
0
votes
1answer
51 views
Could $\sum e^{a_i}$ be simplified? Does it have an identity?
$\sum_{i=1}^n e^{a_i}$ (where $a_i \in \mathbb R$) is expensive for large $n$ (a sum and $n$ exponential operations). I was wondering if there is any way for simplifying this?
2
votes
3answers
77 views
$a^b = c$, is it possible to express $b$ without logarithms?
$ a^b = c $
is it possible to express b without logarithms?
1
vote
1answer
97 views
Raise a number to the “y” power without using exponents.
This is kind of a spinoff on my question Divide by a number without dividing.
Can anyone think of some clever ways to raise any given number to any given power without using an exponent anywhere in ...
1
vote
0answers
35 views
How to solve for the matrix in a set of equations involving the matrix exponential?
I was wondering how to solve the following problem (in a least-squares sense):
$$
\mathbf{y}_1 = e^{Ax_1} \mathbf{y}_0 \\
\mathbf{y}_2 = e^{Ax_2} \mathbf{y}_0 \\
\vdots\\
\mathbf{y}_n = e^{Ax_n} ...
7
votes
5answers
172 views
What does $x^\pi$ mean? [duplicate]
I was just wondering, what does $x^\pi$ or for that matter, $x$ raised to any irrational number mean? For example, I want to represent $x^2$ then that would mean $x * x$ or if I want to do ...
0
votes
0answers
47 views
Prove equation based on square roots
Hi this is my first post,
I am trying to reduce a equation and one of the online lectures and I dont know how he did this:
$$
\exp\left({i k \sqrt{(\xi -x)^2+(\eta -y)^2+z_1^2}}\right)\to \exp({i k ...
10
votes
2answers
129 views
Basic Mathematics. Trouble with proof, powers and odd numbers.
Greets,
In the exercises, at the end of chapter 1.4, Basic Mathematics, Serge Lang
6) Prove: If $n$ is odd, then $\quad (-1)^n = -1$
How?
The working I did
$$\begin{align}( -1)^n &= ( -1 ...
3
votes
0answers
37 views
Order of Recursion?
Define an extended algebraic function f(a) as a function on a that utilizes any combination of recursive extensions and inversions of sequentiation. Example:
a + 1 = sequentiation. a + a = addition ...
3
votes
1answer
44 views
Why is real exponentiation continuous in the base?
I know that real exponentiation is continuous in the exponent ($f(x)=a^x$ is continuous), but how do we know real exponentation is continuous in the base?
What I mean is, if $r$ is an arbitrary real ...
2
votes
4answers
72 views
Basic Mathematics. Trouble with powers and polynomials.
Greets,
I'm hoping to complete the exercises in chapter 1.3 Basic Mathematics, Serge Lang.
The section question is:
Expand the following expressions as sums of powers of $\;x\;$ multiplied by ...
5
votes
6answers
114 views
Basic Mathematics. Trouble with powers.
Greets,
In Chapter 1.3, Basic Mathematics, Serge Lang, there is the question:
Express each of the following expressions in the form $2^m3^na^rb^s$, where $m, n, r, s$ are positive integers.
b) ...
1
vote
2answers
31 views
Do inequalities hold under square-root (or exponentiation in general)?
This has been bothering me lately. My proof-skills are rusty (and were never great to begin with). I dimly recall having seen this (or something related to it) in a math course I took a while ago, but ...
3
votes
6answers
51 views
Motivation for creation of complex exponentiation
I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived?
0
votes
0answers
46 views
Generalization of $e^{t A}$ to $e^{t^{\alpha}A}$
I would like to give a mathematical structure of an operator of the form $(S_{\alpha}(t))_{t \geq 0}$ where $S_{\alpha}(t)=e^{t^{\alpha} A}$ as it was done for the operator $S(t)=e^{t A}$ for ...
8
votes
3answers
264 views
Why is 1 raised to infinity Not defined and not “1” [duplicate]
$1$ square is $1$, so is raised $1$ to $123434234$.
My maths teacher claims that $1$ raised to infinity is not $1$, but not defined. Is there any reason for this?
I know that any number raised to ...
0
votes
1answer
51 views
A calculator's solution to irrational exponent
An irrational number cannot be represented by $\frac{p}{q}$ where $p$ and $q$ are integers.
And when we encounter exponents with decimal points, it is a possible way and a rather simple one to turn ...
1
vote
1answer
71 views
How would I go about solving this type of exponential equations?
I can solve $4^x+5^x=41$:
$4^x=41-5^x$
The left side is strictly increasing, the right side is strictly decreasing. That means that there's at most one solution. That would be 2. Done.
What would I ...
6
votes
6answers
360 views
How is this proof flawed?
$\sqrt{x}=-1$
$\sqrt{x}^2=(-1)^2$
$x=1$
Now substitute it into the original equation
$\sqrt{1}=-1$
$1=-1$
4
votes
1answer
58 views
Simplifying $y=2^{2/3} + 2^{-1/3}$
I am working on a calculus problem where I have to find the local minimum. The value I got was $$y=2^{2/3} + 2^{-1/3}.$$ I simplified it and got this:
$$ y=2^{2/3} + \frac{1}{2^{1/3}}$$
...
2
votes
1answer
33 views
Formula/Algorithn for Exponential factoring?
Given $s = a^b$ find $a$ and $b$. my first algorithm was the obvious brute force method of checking all $b$ roots or dividing by all possible $a$. But I am wondering if there is a more efficient ...
0
votes
1answer
35 views
Finding that probability of the event is small
Let $x_1, \ldots, x_n$ be Bernoulli random variables with the probability of success $P(x_i=1)=p$.
Let $\epsilon>0$.
Show that probability
$$
P\left(\left|\sum_{i=1}^nx_i-p\right|> ...
4
votes
4answers
67 views
Can you raise a Matrix to a non integer number? [duplicate]
So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
2
votes
2answers
57 views
Division with negative exponents
I have a problem that looks like this:
$$\frac{20x^5y^3}{5x^2y^{-4}}$$
Now they said that the "rule" is that when dividing exponents, you bring them on top as a negative like this:
...
9
votes
5answers
367 views
Evaluating tetration to infinite heights (e.g., $2^{2^{2^{2^{.^{.^.}}}}}$)
The Problem
How can you evaluate (i.e., get a value for) Tetration (i.e., iterated exponentiation) to infinite heights?
For example, what would be the value of this expression?
$$ ...
0
votes
1answer
75 views
De Moivre's theorem question.
State De Moivre's theorem and use it to find integers $ A,B,C$ such that $$\sin^5\theta=A\sin5 \theta + B\sin3\theta + C\sin\theta.$$
I know De Moivre's theorem, how to prove it, and converting to ...
0
votes
1answer
31 views
Put the following in rectangular form.
$$(\sqrt{3}+i)^7$$
My question: $r = 2$. For $\theta$, do I use $\dfrac{\pi}{6}$ or $\dfrac{\pi}{6} + 2n\pi$? The book uses the former but I thought the latter is more appropriate.
Thank you.
0
votes
3answers
81 views
Find the smallest natural number that satisfy $13^N = 1 \pmod {2013}$
Moderator Note: This is a current contest question on Brilliant.org.
Find the smallest natural number that satisfy:
$$13^N = 1 \pmod {2013}$$
My idea is to use the Fermat's little theorem ...
0
votes
0answers
21 views
check validity of following manipulation
in my algebra book,there is written following well known identity
$e^{2*\pi*i}=1$
generally we can use also this identity $e^{k*\pi*i}=(-1)^k$
and if instead of $k$,we put $2$ we get ...
2
votes
3answers
59 views
0
votes
1answer
27 views
Exponent, logarithmic question
I'm reading an article related to bioinformatics and I found this formula:
Probability of $x =(1-y/n)^t$ or approximately $e^{-yt/n}$.
My question is how do we pass to the approximation given in the ...
3
votes
4answers
70 views
Proof of $\sqrt{2^{2^k}} = 2^{2^{k-1}}$?
It's quite easy to observe that for $k \ge 0$:
$$
\begin{align}
2^{2^k} &= 4, 16, 256, 65536, \dots\\
\sqrt{2^{2^k}} &= 2, 4, 16, 256,\dots
\end{align}
$$
More in general:
$$
\sqrt{2^{2^k}} ...
3
votes
3answers
42 views
Triplets 4th of power of first equals to sum of other two
I was looking through the admission test for the University Normale of Pisa and I found a problem that I don't know how to solve, it state something like that:
Find all the triplets of number (x, y, ...
10
votes
4answers
711 views
What's the difference between $3^{3^{3^3}}$ and $27^{27}\;$?
Why does $\;\large3^{3^{3^3}}\;$ evaluate to a larger number than $\;\large 27^{27}$?
0
votes
1answer
82 views
For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?
I came across the following question:
For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?
What I have tried:
Since for $ \alpha = 1 $ is clearly non-differentiable in ...
0
votes
1answer
17 views
exponentiation question
I have a homework question on exponents, the question asks "simplify the expression and eliminate any negative exponents"
The Question is as follows
$$(2x^2y^4)^3(3x^{-3}y)^2$$
and my working out
...
0
votes
5answers
167 views
Generate solutions of Quadratic Diophantine Equation
Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: How to solve Quadratic Diophantine Equation
Here's the answer:
$$ ...
0
votes
2answers
68 views
$ 10^{-9}[2\times10^6 + 3^{1000}] $
$$ 10^{-9}[2\times10^6 + 3^{1000}] $$
I'm stuck on solving this. I wasn't able to put this into my calculator since the number is too big for it to calculate.
So far I've done this:
$$ ...
0
votes
2answers
131 views
Raising a square matrix to a negative half power
I want to implement the following formula (taken from Kaiser, 1970) in R where $R$ is square matrix of correlations:
$$S = (\textrm{diag } R^{-1})^{-1/2}$$
I understand the diagonal and inverse ...
3
votes
3answers
174 views
How to solve Quadratic Diophantine Equation
Here's the problem.
Find the solutions of the following equation:
$$ k^2 - 1 = 5(m^2 - 1).$$
Here's my idea:
The original equation can be written as:
$$ k^2 = 5m^2 - 4 \Longleftrightarrow k^2 - ...
9
votes
2answers
257 views
1
vote
3answers
70 views
conditions under which real-matrix exponential are equivalent
Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$?
Thanks!
1
vote
2answers
41 views
Calculating z^n without trigonometric functions
I'm looking for a formula to calculate z^n for complex z and integer n. I know that I can use the Moivre's formula but I will use the formula in a computer program ...
15
votes
6answers
615 views
A question comparing $\pi^e$ to $e^\pi$ [duplicate]
I was doing an algebra problem set following a chapter on logarithms and exponentiation, and it presented this "bonus question":
Without using your calculator, determine which is larger: $e^\pi$ ...
15
votes
4answers
236 views
Intuition for $\omega^\omega$
I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
3
votes
2answers
95 views
infinite derivative of $e^x$
i have started thinking about one topic a few days ago and i am confused if i am wrong or what happens,generally we know that function $e^x$ is somehow 'magic',which means that ...
