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

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2
votes
1answer
26 views

General formula for exponent of a semidirect product

This page states that Exponent of semidirect product may be strictly greater than lcm of exponents But it doesn't give any proof for that. Could anyone provide a general formula for the ...
0
votes
1answer
41 views

Solve $g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$

Let $y$ be a real number. Find $g$ such that $$g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$$ Is valid for all real $x$.
-2
votes
1answer
52 views

Simplifying Exponents (exponent laws) [closed]

So I have this equation: $${\left(\left(\frac{w}{x}\right)^{\frac{y}{x}}\right)}^{\frac{z}{x}}$$ And I know that $${\left(a^b\right)}^c = a^{bc}$$ So I figured I'd simplify it into this: ...
2
votes
1answer
83 views

If $q, r \in \mathbb{R}, x \in \mathbb{R}^+$ then $(x^q)^r=x^{qr}$

I'm stuck on this exercise from Tao's Analysis 1 textbook: show that if $q, r \in \mathbb{R}, x \in \mathbb{R}^+$ then $(x^q)^r=x^{qr}$. DEF. (Exponentiation to a real exponent): Let $x>0$ be ...
2
votes
0answers
42 views

Integral of the product of a power function and an arbitrary exponentiated function

I was only able to find integral tables that solve $$f(t)=\int t^c e^{kt}dt$$ but the integral I'm trying to solve has a function, not a constant, for the exponent: $$f(t)=\int t^c e^{g(t)}dt$$ Is ...
3
votes
3answers
80 views

Is $(-1)^{ab} = (-1)^{ba}$ true? => $(-1)^{ab} = ((-1)^a )^b$ is true? [duplicate]

In general we know $A^{bc} = A^{cb}$ for integer $A$. I want to extend this to the case $A=-1$. For integers $a,b$ I guess the above relation holds, \begin{align} (-1)^{2\cdot3} = ((-1)^2)^3 = 1 = ...
1
vote
1answer
41 views

General formula to compute the exponent of the symmetric group $S_n$

Someone has already asked whether an exponent less than $n!$ is possible for a symmetric group $S_n$. It has been answered that it is for $n \ge 4$. I would like to know if there is a general ...
5
votes
4answers
74 views

Solve the equation: $2^{2x+1}=\left(\frac{1}{32}\right)^x$

Having trouble with this problem: $$2^{2x+1}=\frac{1}{32^x}$$ Do I need to set the exponents equal to each other?
-2
votes
1answer
122 views

Please help me solve real-analysis problem [closed]

Problem: Assume we have the next recursive sequence: $$\begin{cases}x_n=\sqrt[3]{6+x_{n-1}}\\x_1 = \sqrt[3]{6}\end{cases}$$ Prove that there exists a constant $C \neq 0$ such that: ...
3
votes
1answer
44 views

What is $1^\omega$?

In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as: If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, ...
0
votes
0answers
31 views

Find all solutions to $a^b = b^a$ [duplicate]

Find all ordered pairs $(a, b)$ such that $a$ and $b$ satisfy $a^b$ = $b^a$ and $a$ and $b$ are integers. The only way I can think of solving this question is by trial and error, but there must ...
1
vote
1answer
63 views

Why is it justified to move the limit into the exponent?

On my last math test my teacher told me that my notation for evaluating limits "might be problematic." The notation he is referring to is when evaluating a limit of the form $$\lim_{x\to ...
0
votes
0answers
13 views

Bounding an expression

I am trying to figure out an upper bound on the following expression $$(1 + \epsilon)^{\frac{A}{1+\epsilon} - B}$$ where $\epsilon \in (0,1)$, $A \in (0,1)$ and $B \in \{0, 1\}$. I tried doing the ...
0
votes
3answers
52 views

Find the value of the expression $\left (2 + 5 \right ) + \left (2^{2}+5^{2} \right ) + \left (2^{3}+5^{3} \right ) + \left (2^{4}+5^{4} \right )$

How to effectively solve this expression? $$\left (2 + 5 \right ) + \left (2^{2}+5^{2} \right ) + \left (2^{3}+5^{3} \right ) + \left (2^{4}+5^{4} \right )$$ Inefficient method: $$\left ...
0
votes
0answers
41 views

What are the solutions to $n^k \equiv k \mod m$?

Question: For a given modulus $m$ and base $n$, I am examining the set of solutions $\{k \in [0, m) \:\: | \:\: n^k \equiv k \mod m\}$. The case when $m$ is a power of 10 interests me the most. Why ...
3
votes
3answers
89 views

Limits using L'Hopital's rule $\lim_{x\to0^+} (x^{x^x-1})$

Could you help me with this one? Thanks. The answer should be 1, somehow. I tried everything I know, but I couldn't solve it. $$\lim_{x\to0^+} (x^{x^x-1})$$
0
votes
2answers
24 views

Exponential to polar form

I have exponential form $$ je^{-j\pi/2} $$, where $j = \sqrt{-1}$ I want to convert this to polar form $$j(\cos\pi/2 + j \sin \pi/2)$$ is it correct?
1
vote
1answer
66 views

Comparing $3^{1431}$ with $2^{2010}$ without logarithms

I want to compare : $3^{1431}$ and $2^{2010}$ I tried logarithms, $\mathrm{log}_2$ is the way to go. $\mathrm{log}_2(2^{2010})=2010$ $\mathrm{log}_2(3^{1431})=1431\,\log_2(3)=2268.08$ since ...
3
votes
2answers
25 views

Exponential Properties

Here are my steps: $e^{{2\pi i}/100} = (e^{\pi i})^{{2/100}} = ((-1)^2)^{1/100} = 1^{1/100} = 1$. I'm not sure if the normal rules of exponents apply like this if the power is complex.
14
votes
1answer
134 views

Numbers whose powers are almost integers

Some real numbers $\alpha$ have the property that their powers get ever closer to being integers -- more precisely, that $$ \lim_{n\to\infty} \alpha^n-[\alpha^n] = 0 $$ where $[\cdot]$ is the ...
-1
votes
2answers
51 views

$x^y = \exp( \ln(x) \cdot y )$, not a real solution for decimal numbers?

I am trying to understand how to calculate $x^y$ where $y$ is a decimal number, ($2^{2.3}$) According to wikipedia, the 'solution' would be $$ x^y = \exp( \ln(x) \cdot y ).$$ But if we break it ...
1
vote
1answer
33 views

Quick check on multiplied powers

I'm feeling a little silly askign this question, but after about 2 hours of circling around the same point I am getting frustrated. Starting with the expressions for $M$ and $R$ from the lecture ...
1
vote
1answer
19 views

Variables and exponents

How would you solve this equation ? $$500n=4000(1.016)^n$$ I tried using some logarithms but I could not do it. The only unknown variable is n but I'm having a bit of trouble getting there.
8
votes
3answers
118 views

Simplify the expression $(a+1)(a^2+1)(a^4+1)\cdots(a^{32}+1)$

How do I simplify this expression? $$\left( a+1 \right) \left( a^{2}+1 \right) \left( a^{4}+1 \right) \left( a^{8}+1 \right) \left( a^{16}+1 \right) \left( a^{32}+1 \right)$$
1
vote
3answers
302 views

Is there an algorithm to split a number into the sum of powers of 2?

Am I able to split, lets say 76, into the sum of powers of two, through an algorithm and without cycling through possible combinations? For the example above, the answer would be '2^6+2^3+2^2' or ...
0
votes
0answers
15 views

sample size calculation for count data

I have a plan to see some treatment effect in several projects in my company where I will compare the average number of errors now and after the treatment. So what I know from current situation is ...
0
votes
1answer
40 views

Supremum and infimum of powers with rational exponents. [duplicate]

I asked this question earlier and got a very useful hint, but I still can't get the full solution. Here is the original question: I'm trying to show that for $b>1$, $x>0$ and $x$ irrational, ...
1
vote
2answers
43 views

An interesting log problem $3^{{(log_3{x})}^2}+x^{log_3x}=162$

$$3^{{(\log_3{x})}^2}+x^{\log_3x}=162$$ How do I go about doing this. I am stuck at the step $x^{\log_3x} = 81$. Is this right? How do I continue or is it wrong?
3
votes
0answers
37 views

Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$

I'm trying to show that for $b>1$, $x>0$ and $x$ irrational, that $$\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$$ I know this follows immediately if we define ...
2
votes
0answers
36 views

Exponentiation - I can do this?

Euler showed that: $$ e^{i\pi} + 1 = 0 $$ So I thought: $$ b \in \mathbb R,i = \sqrt{-1} \\ n^{bi} = e^{bi\ln n}\\ b = 2k\pi\\ e^{bi\ln n} = e^{2k\pi i\ln n} = (e^{i\pi})^{2k\ln n} = ((-1)^2)^{k\ln ...
0
votes
4answers
59 views

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$ Sorry for the really dumb question but I'd like to see the process of how this is achieved.
1
vote
0answers
64 views

When is $0^0 = 1$ inconvenient? I heard sometimes $0^0 = 0$ may help.

Don't get me wrong: my question is NOT claiming $0^0 = 1$. I understand it's indeterminate. Many articles show that defining $0^0 = 1$ is (just) convenient and I completely agree. However, I've heard ...
0
votes
1answer
24 views

Existance of a nonempty subset $X$ of $\mathbb{Z}$: $\forall x \in X, k \in \mathbb{N}, \exists ! y \in X$ satisfying $\mid x-y \mid 2^k$

Define the set $A = \{2^k \mid k \in \mathbb{Z}_+\}$ Does there exist a nonempty subset $X$ of $\mathbb{Z}$ satisfying the following condition: For all $a \in A$ and $x \in X$, there exists a unique ...
-2
votes
2answers
50 views

finding the second smallest number [closed]

Please tell me how to solve this kind of problem in a fast manner. Which of the following is the second smallest number: $2^{120}$, $3^{80}$ and $10^{30}$? Hope you could show me the best solution, ...
4
votes
2answers
131 views

Which general physical transformation to the number space does exponentiation represent?

Addition and multiplication may be defined in two ways, one specific and one general: Addition specific: addition is repeated incrementation. This is specific and sub-optimal as while $2 + 4$ is ...
4
votes
3answers
50 views

The value of the cubic root of $-i$

So this was the question given to us. $\left(\iota=\sqrt{-1}\right)$ Value(s) of $\left(-\iota\right)^{\dfrac{1}{3}}$ are (A) $\dfrac{\sqrt{3}-\iota}{2}$ (B) $\dfrac{\sqrt{3}+\iota}{2}$ ...
0
votes
2answers
43 views

If $a^{(b^c)}=d^c$, find $d$ in terms of $a$ and $b$.

Is it possible to express $d$ in terms of $a$ and $b$ only in the following equation? $$a^{b^c}=a^{(b^c)}=d^c$$ I want something like $d=\dots$ Thanks in advance!
2
votes
1answer
106 views

Exponential Factorial vs Tetration

I'm wondering whether there's a known way to compare the exponential factorial of n versus the tetration of a fixed number $($ e.g., $3$, since it appears in Graham's number $)$ with the same number ...
1
vote
0answers
21 views

Solving an Equation Involving Modular Exponation

Is it possible to solve the following equation? $$ 3^k\mod k \equiv 24 $$ Clearly $k>24$; does a solution definitely exist, and can it be found in any simple way? I can certainly calculate ...
2
votes
2answers
23 views

Find real constants $c$ and $k$ such that $y=cx^k$ passes through point $(a, b)$ with slope $m$

In the Cartesian plane, can a power function of the form $y=cx^k$ (where $c>0$ and $k>1$, not necessarily an integer) be found such that its graph passes through any arbitrary point $(a, b)$ ...
0
votes
2answers
39 views

Negative exponents on a quantity in scientific notation considering significant figures

Are there rules that apply to negative exponents with regard to scientific notation? The specific problem is: $$\left(6.3\times10^{2}\right)^{-6}$$ I believe the following is correct: ...
1
vote
1answer
44 views

Convert $3^n$ to some form of $2^n$

I am not from Maths field, but I need your help to convert the $3^n$ form to $2^n$. I need to change the base from $3$ to $2$. The resultant expression can be of any form.
4
votes
1answer
71 views

Why is $\gcd(2^p + 1,3^p + 1) = 1$?

Let $p$ be an odd prime. Why is $\gcd(2^p + 1,3^p + 1) = 1$ ? I tried using fermat's little and $\gcd(a+b,a) = gcd(a,b)$ but without succes. I can make a statistical argument that suggests there are ...
3
votes
1answer
49 views

Number of integer triples to exponential equation

I'm taking a class on number theory and this is one of the problems my professor gave. How many ordered integer triples $(x,y,z)$ are there such that $x^y-y^x=2017\times z$, where $x,y$ are less than ...
0
votes
1answer
43 views

Find the number of times we can take log base two of this exponent

I'm not sure how to simplify this type of exponential expression. I would like to know $k$ many $log_{2}^{k}(n)$ such that $n \leq 1$ $$n = (2^{2^{2^{15}}})^2$$ So attempt I to simplify $$n = ...
0
votes
1answer
39 views

Generalized Complex Exponentiation

Is there a way to create a general formula for exponentiating two complex numbers like there is for addition and multiplication? Ex: $(a + bi) + (c + di) = (a + c) + (b + d)i $ $(a + bi) * (c + di) ...
1
vote
2answers
128 views

Simplifying algebraic fraction, exponents

Would someone be able to tell me how $$\bigg( \frac{5}{a^4} \bigg)^{-3}$$ gets simplified to $$\frac{a^{12}}{125}?$$ Thank you!
0
votes
1answer
23 views

How to simplify algebraic expression with exponents?

Can someone please tell me step-by-step how to simply this? $$\sqrt{\frac{8x^{{\frac{1}{2}}^{\frac{2}{3}}}}{x^{-\frac{1}{2}}}}$$ Edit: correct answer is 2x^(5/12), I'm just not sure how to do it. ...
1
vote
4answers
86 views

How do you prove that for all real numbers c, there always exists a value n, in which $4^n > c3^n$?

What we have so far, its $\frac{4^n}{ 3^n} > c,$ such that $\left(\frac{4}{3}\right) ^ n > c.$ From that I know that it is true, but I don't really know how to prove it formally.
2
votes
2answers
79 views

I am not sure, but it's an equation…

$$2x + 2^x = 4$$ So it's clear that $x$ will equal $1$ but how can it be solved through an algebraic method to be able to determine $x$ with more complex numbers. I've tried to transform the exponent ...