Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

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206
votes
23answers
17k 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 \cdot 0^x = 1 \cdot ...
19
votes
4answers
3k views

How do I compute $a^b\,\bmod c$ by hand?

How do I efficiently compute $a^b\,\bmod c$: When $b$ is huge, for instance $5^{844325}\,\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, ...
55
votes
6answers
6k 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 $x^y = y^x$?
49
votes
9answers
4k views

Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
18
votes
4answers
6k views

Finding $\int x^xdx$

I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$. $$\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] ...
47
votes
5answers
13k views

Prove that $i^i$ is a real number

According to WolframAlpha, $i^i=e^{-\pi/2}$ but I don't know how I can prove it.
14
votes
2answers
999 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 $x$?
38
votes
3answers
2k views

What is the value of $1^i$?

What is the value of $1^i$? $\,$
16
votes
1answer
773 views

Prove that $(p+q)^m \leq p^m+q^m$

If $p,q$ are positive quantities and $0 \leq m\leq 1$ then Prove that $$(p+q)^m \leq p^m+q^m$$ Trial: For $m=0$, $(p+q)^0=1 < 2= p^0+q^0$ and for $m=1$, $(p+q)^1=p+q =p^1+q^1$. So, For ...
68
votes
4answers
4k views

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

Find $x$ in $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$ A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = ...
13
votes
10answers
1k 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 ...
75
votes
15answers
5k views

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
129
votes
9answers
5k 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 ...
15
votes
6answers
15k 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 ...
1
vote
5answers
509 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: $$ ...
27
votes
1answer
3k 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 ...
16
votes
1answer
467 views

If $n^c\in\mathbb N$ for every $n\in\mathbb N$, then $c$ is a non-negative integer?

Supposing that a real number $c$ is given, is the following true? "If $n^c$ is a natural number for every natural number $n$, then $c$ is a non-negative integer." Though this seems true, I can't ...
14
votes
3answers
2k views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
5
votes
3answers
5k 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.
20
votes
18answers
2k views

How to understand why $x^0 = 1$, where $x$ is any real number?

Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$ I understand that the ...
8
votes
4answers
4k 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 ...
52
votes
10answers
3k 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 ...
55
votes
2answers
3k views

Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
35
votes
12answers
4k 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 ...
21
votes
5answers
4k 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 ...
26
votes
9answers
2k 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.
5
votes
2answers
2k 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} & ...
37
votes
4answers
12k 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. ...
6
votes
5answers
236 views

Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger?

Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger? I tried an induction approach. First I showed that if $b=3$ then any $a \geq4$ satisfied $a^b<b^a$. Then using that ...
20
votes
6answers
6k 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.
11
votes
5answers
23k views

How do you calculate the modulo of a high-raised number?

I need some help with this problem: $$439^{233} \mod 713$$ I can't calculate $439^{223}$ since it's a very big number, there must be a way to do this. Thanks.
10
votes
4answers
668 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} ...
14
votes
4answers
6k 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.
18
votes
6answers
7k 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: $e^\pi$ ...
7
votes
1answer
164 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
7
votes
2answers
266 views

An “elementary” approach to complex exponents?

Is there any way to extend the elementary definition of powers to the case of complex numbers? By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots ...
4
votes
2answers
4k views

Proving the Product Rule for exponents with the same base

For all $ a, b \text{ and } c \in \mathbb{R}$ and $a>1$, Prove that $a^b\cdot a^c=a^{b+c}$ I have come across this question and its bugging me. Its a basic property that we learn in HS and I was ...
5
votes
3answers
5k views

$x^x=y$. How to solve for $x$? [duplicate]

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
4answers
928 views

What is the value of $i^i$? [duplicate]

I understand that when you raise any number $x$ to a power, you multiply $x$ by itself the number of times indicated in the power. However, what happens when $i^i$ is performed? How can a number be ...
4
votes
8answers
758 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 ...
1
vote
2answers
174 views

Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$

Hi guys this was a practice problem I was given, can anyone help me out on it? This is the problem: Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$ and the following is what I ...
118
votes
3answers
10k 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. ...
26
votes
10answers
819 views

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
37
votes
1answer
867 views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
20
votes
6answers
51k views

How to calculate a decimal power of a number

I wish to calculate a power like $$2.14 ^ {2.14}$$ When I ask my calculator to do it, I just get an answer, but I want to see the calculation. So my question is, how to calculate this with a pen, ...
4
votes
1answer
189 views

Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

Conjecture: There does not exist a non-integer $x$ such that $$2^x=a$$ $$3^x=b$$ $$5^x=c$$ where $a,b,c$ are all integers. I'm aware that the similar question There does not ...
23
votes
8answers
900 views

Infinite powering by $i$ [duplicate]

Find the value of: $i^{i^{i^{i^{i^{i^{....\infty}}}}}}$ Simply infinite powering by i's and the limiting value. Thank you for the help.
14
votes
1answer
501 views

How to raise a number to a quaternion power

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: ...
9
votes
2answers
432 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
8
votes
5answers
8k views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...