Questions about exponentiation

learn more… | top users | synonyms (1)

84
votes
14answers
6k views

Zero to the zero power - Is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...
31
votes
5answers
2k views

$x^y = y^x$ for integers $x$ and $y$

We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
24
votes
3answers
1k views

What is the value of $1^i$?

What is the value of $1^i$? $\,$
48
votes
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, ...
7
votes
2answers
605 views

Is $x^x=y$ solvable for $x$?

Given that $x^x = y$; and given some value for $y$ is there a way to expressly solve that equation for $x$?
6
votes
4answers
2k views

Non-integer powers of negative numbers

Roots behave strangely over complex numbers. Given this, how do non-integer powers behave over negative numbers? More specifically: Can we define fractional powers such as (-2)^-1.5? Can we define ...
5
votes
3answers
2k views

Inverse of $y=xe^x$

I feel like finding the inverse of $y=xe^x$ should have an easy answer but can't find it.
94
votes
9answers
3k views

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
20
votes
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 ...
10
votes
4answers
516 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} ...
22
votes
3answers
4k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
14
votes
5answers
2k views

$\sin(A)$, where $A$ is a matrix

If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
15
votes
5answers
2k views

Approximation of $e^{-x}$

Is there a method to mentally evaluate $e^{-x}$ for $x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters.
3
votes
2answers
326 views

Why does the $n$-th power of a Jordan matrix involve the binomial coefficient?

I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$-th power is: $$ J_k(\lambda)^n = \begin{bmatrix} \lambda^n & \binom{n}{1}\lambda^{n-1} & ...
101
votes
3answers
7k views

Is 2048 the highest power of 2 with all even digits (base ten)?

I have a friend who turned 32 recently. She has an obsessive compulsive disdain for odd numbers, so I pointed out that being 32 was pretty good since not only is it even, it also has no odd factors. ...
21
votes
6answers
2k views

Why do we need to prove $e^{u+v} = e^ue^v$?

In this book I'm using the author seems to feel a need to prove $e^{u+v} = e^ue^v$ By $\ln(e^{u+v}) = u + v = \ln(e^u) + \ln(e^v) = \ln(e^u e^v)$ Hence $e^{u+v} = e^u e^v$ But we know from basic ...
12
votes
4answers
3k views

How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?

How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$? When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise? Thanks.
14
votes
11answers
931 views

Alternative notation for exponents, logs and roots?

If we have $$ x^y = z $$ then we know that $$ \sqrt[y]{z} = x $$ and $$ \log_x{z} = y .$$ As a visually-oriented person I have often been dismayed that the symbols for these three operators ...
16
votes
4answers
437 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 ...
8
votes
3answers
3k views

How do you compute negative numbers to fractional powers?

My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure ...
3
votes
3answers
586 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 - ...
0
votes
5answers
304 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: $$ ...
19
votes
9answers
1k views

Why is $x^0 = 1$ except when $x = 0$?

Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
3
votes
3answers
1k views

Complex Exponents

What does it mean to raise a number to a complex exponent, and why? A lot of the explanations that I've seen involve e, why is this? I'm looking for an intuitive answer describing to how ...
9
votes
10answers
311 views

Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$

I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no ...
4
votes
7answers
560 views

Numbers to the Power of Zero

I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations. This is a three part question, the parts are as ...
2
votes
5answers
1k views

Why is two to the power of zero equal to binary one?

Probably a simple question and possibly not asked very well. What I want to know is.. In binary, a decimal value of 1 is also 1. It can be expressed as $x = 1 \times 2^0$ Question: Why is two to ...
41
votes
9answers
2k views

What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
15
votes
1answer
447 views

How to prove $\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\frac{\pi-1}{2}$

One of my classmates challenged me to solve $\displaystyle\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\;?$ With a simple c program I found that $\displaystyle\sum\limits_{n=1}^{1048576}\frac{\sin ...
31
votes
14answers
3k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
16
votes
6answers
2k 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: $e^\pi$ ...
20
votes
5answers
2k views

Fun logarithm question

How would you go about solving $x^{2^{x}} = 2^{x}$? There should be a solution $1<x<2$, but I haven't found a way to derive the answer using the usual log laws, maybe there is an elegant way ...
7
votes
4answers
433 views

Please help me to show, that $(\ln x)'=\frac1 x$

In school, we recently started with derivations. I looked into a list of simple derivations and tried to prove them, in order to practice. Now, I tried to find the derivative of $\ln x$, but I got ...
6
votes
6answers
210 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
9
votes
2answers
288 views

Algebraic equation problem - finding $x$

$$(x^2 +100)^2 =(x^3 -100)^3$$ How to solve it?
9
votes
1answer
401 views

What is wrong with this funny proof that 2 = 4 using infinite exponentiation?

Out of boredom, I decided to recall the following equation: $$x^{x^{x\cdots}} = 2.$$ Which, I simply rewrote like this: $x^2 = 2$, and therefore $x = \sqrt{2}$. Then I took a look at the more ...
5
votes
2answers
97 views

Extending exponentiation to reals

I've been reading through a course on exponential functions, starting from integer-valued exponents to rational ones as in: $x^r$ from $r\in \Bbb{N}$ to $\Bbb{Z}$, and combining them to rigorously ...
5
votes
1answer
900 views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
4
votes
2answers
170 views

How to evaluate powers of powers (i.e. $2^3^4$) in absence of parentheses?

If you look at $2^{3^4}$, what is the expected result? Should it be read as $2^{(3^4)}$ or $(2^3)^4$? Normally I would use parentheses to make the meaning clear, but if none are shown, what would you ...
1
vote
1answer
460 views

Why can't you integrate all power functions without a log function?

You need a logarithm function to solve all power functions. That's a fact. Power functions look like this: $f\colon x \mapsto a x^r \qquad a,r \in \mathbb{R}$ But why would you need a logarithm ...
9
votes
1answer
770 views

Matrix raised to a matrix

Good evening, I was wondering if there is such a valid operation as raising a matrix to the power of a matrix, e.g. vaguely, if $M$ is a matrix, is $$ M^M $$ valid, or is there at least something ...
8
votes
5answers
399 views

Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}$

I'm so puzzled about this: $$a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}.$$ Why isn't $a^{b^c}$ equal to $a^{(bc)}$? Why is $a^{b^c}$ instead equal to $a^{(b^c)}$? And how is it possible that ...
6
votes
7answers
234 views

exponential equation

$$\sqrt{(5+2\sqrt6)^x}+\sqrt{(5-2\sqrt6)^x}=10$$ So I have squared both sides and got: $$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2\sqrt{1^x}=100$$ $$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2=100$$ I don't know what to do ...
5
votes
2answers
2k views

How can I calculate non-integer exponents?

I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function: $$ f(x,y) = \begin {cases} 1 & y = 0 \\ (x)f(x, y-1) & y > 0 \end ...
4
votes
1answer
242 views

How to solve infinite repeating exponents

How do you approach a problem like (solve for $x$): $$x^{x^{x^{x^{...}}}}=2$$ Also, I have no idea what to tag this as. Thanks for any help.
4
votes
3answers
2k views

$x^x=y$. How to solve for $x$?

I tried looking for ways to solve this equation and came across something like Lambert's W function, which, by the way, I did not understand a bit, because I've never learned it nor do I have a decent ...
3
votes
3answers
172 views

$x^{y^z}$: is it $x^{(y^z)}$ or $(x^y)^z$?

Of the following, why is a usually considered true, and for what reason other than "tradition" and "more convenient"? a: ${x}^{y^z} = x^{(y^z)} \neq {(x^y)}^z$ b: ${x}^{y^z} = {(x^y)}^z \neq ...
2
votes
2answers
74 views

How hard is finding values such that

We can work with powers of some naturals $(x_k)^{m_k}$. Here we have $n$ naturals, and $m_k$ is an integer in the range $-r$ to $r$. My question is, how small can $p$ be so that ...
2
votes
4answers
6k views

Modulo arithmetic with big numbers?

I need to calculate $3781^{23947} \pmod{31847}$. Does anyone know how to do it? I can't do it with regular calculators since the numbers are too big. Is there any other easy solutions? Thanks
1
vote
2answers
109 views

Calculating new vector positions

I'm using the following formula to calculate the new vector positions for each point selected, I loop through each point selected and get the $(X_i,Y_i,Z_i)$ values, I also get the center values of ...