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

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0
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3answers
77 views

Which of the folowing no. is largest [closed]

Which of the folowing no. is largest-- $2^{3^{4}}$ , $2^{4^{3}}$ , $3^{2^{4}}$ , $3^{4^{2}}$ , $4^{2^{3}}$ , $4^{3^{2}}$. I am stuck on this problem. Can anyone help me please...
4
votes
2answers
120 views

Characterizing continuous exponential functions for a topological field

Given a topological field $K$ that admits a non-trivial continuous exponential function $E$, must every non-trivial continuous exponential function $E'$ on $K$ be of the form $E'(x)=E(r\sigma (x))$ ...
2
votes
3answers
107 views

Why does $(-2^2)^3$ equal $-64$ and not $64$?

The title says it all. Why does $(-2^2)^3$ equal $-64$ and not $64$? This was on my algebra final, and I am completely stuck on how it works.
6
votes
0answers
216 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
0
votes
2answers
77 views

Square roots and powers

This is a rather silly question. In what order does one evaluate a combination of powers and fractional powers? I have the question phrased: $\sqrt{ 1/4^2 }$ OR ${ 1/4 }$? Which is greater? I ...
2
votes
1answer
54 views

Derivative of $x\times n - 2^{\log_2 {x \times n}}$

I have a problem with solving derivative of $f(x)$ in this case: $$f(x) = x\times 10^9 - 2^{\log_{10} x\times 10^9}$$ This is what I have: $$f^\prime(x) = \lim_{m\to0} {f(x+m) - f(x)\over m}$$ $$= ...
1
vote
1answer
32 views

Using exponent laws to simplify $2^t\times4^2+8^{t-2}$

This is one of the questions in my maths text book and I don't quite get it, can someone please explain it and show the working out? use index laws to find $2^t\times4^2+8^{t-2}$ Observation ...
1
vote
3answers
155 views

solving the equation$(((e^x)^{e^x})^{e^x})^{…}=2$ for x

Let $ \left(x_n\right)_{n = 1}^\infty $ be a sequence of real numbers defined by the initial value $ x_1 = e^x $ for some $ x \in \mathbb{R} $ and the relationship $ x_{n+1} = e^{x^{x_n}} $, such ...
0
votes
3answers
169 views

$1500=P \times { (1 + 0.02) }^{ 24 }$, what is the value of $P$?

Hey guys could you please tell me what is the faster why to solve this equation. It's a compound interest equation and I'm stuck at the ${ (1 + 0.02) }^{ 24 }$ I really don't know how to proceed in ...
6
votes
3answers
221 views

If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?

Is the following true if $\theta\in\mathbb{Q}$? $$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$ Is it true if $\alpha\in\mathbb{R}$? In each case, prove or give a ...
1
vote
2answers
94 views

Do any issues arise if we try to raise an element of $\mathbb{R}^+$ to an element of $\mathbb{C}$?

If $a$ and $b$ are non-zero natural numbers, the definition of $a^b$ is clear. Now it seems to me that there are (at least) two distinct ways of generalizing to larger number systems. Firstly, given ...
1
vote
1answer
123 views

I just found out that $0^0$ equals $1$, why is this? [duplicate]

I have done a lot of math so far, but I never stumbled on something this simple and yet mind boggling. Can someone tell me why $0^0$ equals $1$? I always knew that everything raised to a power of $0$ ...
0
votes
1answer
148 views

k-fold matrix product

For $k \in \mathbb{N}$, $B,C \in \mathbb{R^{n,n}}$, given the matrices $B,C$ , calculate all powers $B^k$ and $C^k$ I'm a bit puzzled by this task. I assume it's supposed to practice handling ...
2
votes
1answer
116 views

Exponential equations involving natural numbers at power “x”

Find x : $$4^x+15^x=9^x+10^x(2^x-3^x)(2^x-3^x-5^x)$$
4
votes
1answer
275 views

Digits in a large power of two

I am trying to find the answer to: 2^34359738368. As to be expected every calculator and computer program I have used has crashed. To be honest I don't even want to know the exact answer, I just want ...
3
votes
2answers
295 views

Zeta function and probability

I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function) But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the ...
4
votes
2answers
132 views

Proof for $\displaystyle\sum_{k=1}^n k^a$ equaling a sum of fractions

I know $\displaystyle\sum_{k=1}^n k^2$ equals $n/6+n^2/2+n^3/3$, but... why? And I also know that $\displaystyle\sum_{k=1}^n k^3$ equals $n^2/4+n^3/2+n^4/4$, but... is there a pattern so I can easily ...
1
vote
2answers
124 views

Is $a^n$ an exponent of $a$?

I see that definition says that $n$ is an exponent. But, the name of the function is normally expanded to the results computed by the function. That is, if we raise $a$ into power $n$ we say that ...
2
votes
1answer
124 views

Proof of correctness of Putzers algorithm

I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into ...
1
vote
3answers
190 views

Integrating $x^x$ and getting a graph [duplicate]

I've heard many times of functions that cannot be integrated. For example, $x^x$, which is the most common. But what I don't know is how could you, even if the graph has no equation, plot this ...
6
votes
6answers
324 views

How can $4^x = 4^{400}+4^{400}+4^{400}+4^{400}$ have the solution $x=401$?

How can $4^x = 4^{400} + 4^{400} + 4^{400} + 4^{400}$ have the solution $x = 401$? Can someone explain to me how this works in a simple way?
7
votes
2answers
163 views

Power tower inequality

I want to prove the following power tower inequality: $$ 3 \uparrow \uparrow 100 > 4 \uparrow \uparrow 99 $$ but I don't know how to do this. I think that induction will not work, because I think ...
5
votes
1answer
65 views

Do we really know the value of expressions with irrational powers?

The way we evaluate decimal powers such as $a^.75$ is by splitting it into $(a^3)$^(1/4). How then can we evaluate irrational powers? I know that we can approximate, but whenever we graph a^x we ...
15
votes
1answer
645 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 ...
2
votes
0answers
1k 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
73 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
158 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.
17
votes
1answer
898 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
110 views

Double summation including power and factorial [duplicate]

I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac {y^m}{m!}$$ Could you please provide a result? Thanks in advance
1
vote
1answer
117 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
69 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
97 views

$a^b = c$, is it possible to express $b$ without logarithms?

$ a^b = c $ is it possible to express b without logarithms?
2
votes
2answers
290 views

rounding up to nearest square

Say I have x and want to round it up to the nearest square. How might I do that in a constant time manner? ie. $2^2$ is 4 and $3^2$ is 9. So I want a formula whereby f(x) = 9 when x is 5, 6, 7 or 8. ...
1
vote
1answer
174 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
73 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} ...
6
votes
5answers
278 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 ...
13
votes
6answers
2k 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 ...
10
votes
2answers
227 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 ...
4
votes
1answer
118 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
99 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
140 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) ...
3
votes
1answer
486 views

taking the log of $a^b$ (Project Euler problem 29)

I've been stuck on Project Euler problem 29 and thus asked a friend who solved it how to do it. What he basically did was for each power was: $\left(\frac{\log_{10}(a)}{\log_{10}(2)}\right)\cdot b$ ...
3
votes
2answers
693 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
0answers
136 views

Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia $$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k ...
1
vote
1answer
78 views

Solving an equation with $x$ as powers

How would I go about solving $$2^x -2^{x-2}=3 *2^{13}$$Hints please. Thank you.
3
votes
6answers
156 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?
2
votes
3answers
77 views

why if x in 1/n power >(<) y in 1/m power then x in c/n power >(<) y in c/m power?

As you might guess this is one more stupid question from non-matematician, and you are right. I found this exercise in "Algebra and trigonometry book": $7^{1/2}$ or $4^{1/4}$. After some googling I ...
8
votes
3answers
2k 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
389 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 ...
6
votes
6answers
494 views

How is this proof flawed?

$\sqrt{x}=-1$ $\sqrt{x}^2=(-1)^2$ $x=1$ Now substitute it into the original equation $\sqrt{1}=-1$ $1=-1$