Questions about exponentiation

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102
votes
3answers
8k 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. ...
101
votes
9answers
4k 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 ...
100
votes
14answers
7k 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 ...
66
votes
5answers
2k views

Root Calculation by Hand

Is it possible to calculate and find the solution of $ \; \large{105^{1/5}} \; $ without using a calculator? Could someone show me how to do that, please? Well, when I use a Casio scientific ...
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}}}}} < ...
49
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, ...
44
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 ...
41
votes
10answers
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 ...
41
votes
2answers
2k 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
11answers
3k views

How is $e^x$ read aloud?

My current research colleague from New Castle told me that I was reading it wrong. I usually read it as e power x. How do you read aloud $e ^ x$? Is it: e raised to x e power x e powered x or e ...
34
votes
6answers
3k 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 $x^y = y^x$?
32
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 ...
29
votes
1answer
551 views

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
27
votes
14answers
2k views

How do I understand $e^i$ which is so common?

Raising something to an imaginary number is weird, I have a hard time wrapping my head around that. And e seems even more common and comes up in many situations, such as: the non-geometric ...
27
votes
3answers
1k views

What is the value of $1^i$?

What is the value of $1^i$? $\,$
24
votes
1answer
351 views

Iterated exponent of $i$

WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
23
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 ...
22
votes
3answers
5k 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. ...
21
votes
10answers
4k views

How to solve $x^{1/2}-x^{1/3} = 0$

How can I solve the following equation? I really can't figure out how to solve it: $x^{1/2}-x^{1/3} = 0$ Thank you.
21
votes
8answers
2k views

How does the exponent of a function effect the result?

The $x^{2/2}$ can be represented by these ways: $$\begin{align} x^{2\over2}=\sqrt{x^2} = |x|\\ \end{align} $$ And $$\begin{align} x^{2\over2}=x^{1} = x\\ \end{align} $$ Which one is correct? And what ...
21
votes
2answers
2k views

Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question. Now, his ...
21
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 ...
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 ...
20
votes
5answers
3k 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 ...
18
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.
18
votes
4answers
304 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
17
votes
7answers
971 views

Simple proof that $8\left(\frac{9}{10}\right)^8 > 1$

This question is motivated by a step in the proof given here. $\begin{align*} 8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\ &=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\ &\geq ...
17
votes
4answers
485 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 ...
17
votes
6answers
1k views

$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$

$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$
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: $e^\pi$ ...
16
votes
5answers
764 views

Do equal bases imply equal powers?

$$x^a = x^b \Rightarrow a =b$$ So, this is a concept I used in multiple math problems and they often turn out right. The thing is, today my math teacher told me that this is not necessarily true. ...
16
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.
16
votes
2answers
257 views

$2^x - a$ touches $\log_2(x)$

I was playing around with the functions $2^x$ and $\log_2(x)$. As they are the inversions of each other, I thought there was a simple number $a$ for which $2^x - a$ touches $\log_2(x)$. Using ...
15
votes
1answer
681 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, ...
15
votes
2answers
348 views

Prove that for any nonnegative integer n the number $5^{5^{n+1}} + 5^{5 ^n} + 1$ is not prime

My math teacher gave us problems to work on proofs, but this problem has been driving me crazy. I tried to factor or find patterns in the numbers and all I can come up with is that for $n > 0$, the ...
15
votes
1answer
504 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 ...
14
votes
11answers
1k 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 ...
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 ...
13
votes
6answers
1k views

10 to the power of 3.5: $10^{3.5}$

So $10^3 = 10\times 10\times 10 = 1000$, this is really easy to understand. But what about: $\,10^{3.5}\,?\,$ My logic would suggest this was $10\times 10 \times 10\times 5 = 5000,\;$ but the ...
13
votes
5answers
427 views

Value of $(-1)^x$ for $x$ irrational

I was working on an analysis problem when this question arose in one my proofs. I think it may be either $-1$ or $1$, but it seems like there can only be an arbitrary way to assign this. So is there ...
13
votes
2answers
1k views

How to know if a number is a power of $x$

I couldn't find anything on the Internet which could direct me to the solution of the following problem. I want to know if $n$ can be calculated by $x^y$ where $y\ge 2$ and $x\ge 2$. I tried using ...
13
votes
6answers
17k 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, ...
13
votes
1answer
181 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: ...
12
votes
6answers
1k views

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

How do I prove the inequality $e^{-2x}\leq1-x$ for $0\leq x\leq1/2$?
12
votes
9answers
4k views

How to calculate $e^x$ with a standard calculator

Is there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...
12
votes
7answers
347 views

integral of $x^2e^{-x^2}~dx$ from $-\infty$ to $+\infty$

I know that the $$\int^{+\infty}_{-\infty}e^{-x^2}~dx$$ is equal to $\sqrt\pi$ It's also very clear that $$\int^{+\infty}_{-\infty}xe^{-x^2}~dx$$ is equal to 0; However, I cannot manage to ...
12
votes
2answers
1k views

Why are the first few powers of $2^{10}$ a little more than those of 1000?

See the complete list here: http://en.wikipedia.org/wiki/Power_of_two#Powers_of_1024. I'm wondering if there's a mathematical explanation for the relationship or if it's just coincidence.
12
votes
3answers
1k views

evaluate the last digit of $7^{7^{7^{7^{7}}}}$

I found this puzzle online. Since I'm not good at number theoretic kind of problems I'm going to propose it in this form. If you have a number $x$, in this case $x=7$, how do you evaluate the last ...
12
votes
4answers
4k 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.
12
votes
4answers
179 views

Under what conditions does $x^{\frac{b}{c}} = (x^b)^\frac{1}{c}$ hold?

It is very common to use the formula $$x^{\frac{b}{c}} = (x^b)^\frac{1}{c}$$ to simplify the evaluation of a fractional exponent. I want to know what circumstances allow us to do this step. For ...