Questions tagged [exponentiation]

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

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2
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1answer
41 views

Showing the inequality $f(x)=x^{2(1-x)}+(1-x)^{2x}\leq 1$ for $0<x<1$

My proof is not really natural but I think it works. We want to show [1]: Let $0<x<1$ such that then we have : $$f(x)=x^{2(1-x)}+(1-x)^{2x}\leq 1$$ Case $0<x\leq 0.25$ The proof of this ...
2
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2answers
88 views

If $A+B \mid A^5$ and $A+B \mid B^5$, what can be concluded?

Assume $A+B \mid A^5$ and $A+B \mid B^5$, while all the variables are integers expect zero. Can we prove that $A=B$? This is my idea of the proof: For every prime $p$ that $p\mid A+B$ there is $p \...
0
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1answer
11 views

Proving the Distributive property of exponents and radicals using bounds $X^(1/n)$

Baby Rudin, 2nd edition, chapter 1, exercise 4 Prove for positive x,y, and positive integer n $\sqrt[n]{x}\sqrt[n]{y}=\sqrt[n]{xy}$ Doing this through induction on n seems reasonable enough (first ...
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2answers
42 views

Extraneous solutions to $i^{2/3}$

I want to find the value of $$i^{2/3}$$ Here was what I tried: $$i^{2/3} = (i^{2})^{1/3} = -1^{1/3} = (-1^{2})^{1/6} = 1$$ I know that I could have also stopped at the third step, since $$-1^{1/3} = -...
0
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1answer
25 views

With what exponent rules does $-(2^{-55} + .4\times2^{-56})+(2^{-54})$ become $.1\times2^{-52}$?

I have a math class focusing on numerical analysis so I'm working with very small numbers. My professor has set $0.4 − 2^{−55} − 0.4×2^{−56} + 2^{−54} = 0.4 + 0.1×2^{-52}$ but shown no steps in ...
0
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1answer
27 views

Previously adding restrictions

Okay, being completely honest, i don't know how more to make it clearer, this question has been deleted 3 times, maybe people don't actually read what i say at the beginning, which said perfectly what ...
0
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0answers
24 views

Is $(\ln{x})^a$ written as $\ln^a{x}$? [duplicate]

This is a basic notation question. With trig functions like $\sin{x}$ and $\cos{x}$, if they are raised to a power, let's say $a$, then we write it, in the case of $\sin{x}$, as $\sin^a{x}$. However, ...
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0answers
33 views

Adding the restrictions [closed]

All i need to know is how to add restrictions to this all these laws (below). I've definitely tried my best to add them and i will post the code of the laws with my attemp at the restrictions, but ...
2
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1answer
46 views

What can be a generalization of repeats in exponentiation using modulo?

I came across a Math Problem in a Japanese Coding Test(It is officially over now so no worries about discussing it, https://atcoder.jp/contests/abc179/tasks/abc179_e). I will write the mathematical ...
4
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3answers
58 views

Restrictions on laws

I'm wondering about the restrictions, my doubt is for example at $\log_a(b)=c\implies a^c=b$, how would anyone add the restrictions for this? I know the argument and the base of a log have to be >0 ...
1
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3answers
47 views

Finding roots : $x-7\sqrt{x}+10$

Find all possible roots of $k(x) = x-7\sqrt{x}+10$ I am having serious trouble rearranging this function as $ax^2+bx+c$ since it has $'-7\sqrt{x}'$. Can anyone please help me? Little help would ...
0
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1answer
27 views

complex to the power rational

I have problem in resolving whether $z^{m/n}=(z^m)^{1/n}$ or $z^{m/n}=(z^{1/n})^m$. For instance, $(-1)^{1/2}=[(-1)^{1/6}]^3$ but $[(-1)^3]^{1/6}=(-1)^{1/6}\neq (-1)^{1/2}$. I find that $(z^{1/n})^m\...
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1answer
33 views

How many days will I need to finish a book if I read more pages every day? [closed]

Lets say a book has 300 pages and I read 10 pages a day + 5% more pages every day ...
1
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2answers
57 views

How to solve $23^{{2020}^{2020}} \bmod 37$? Please see the body of the question.

How to solve $$23^{{2020}^{2020}} \mod 37.$$ Below given is my understanding of trying to solve the problem. From $$x^{p-1} = 1 \mod p$$ I deduce that $$23^{2020} \mod 37$$ would be $$23^{56.36+4} \...
2
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1answer
56 views

Exponential Power Tower

My question is- $$(4x)^{{{{\sqrt x}^{\sqrt x}}^ \cdots}^\infty}=0.0625$$ How to solve it? Options- (A)$2^{1/24}$ (B)$2^{1/48}$ (C)$4^{1/48}$ (D)$2^{1/96}$ I am confused how to solve this infinite ...
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2answers
26 views

Could someone explain to me the property of exponentials and logarithms? [closed]

Why $e^{\ln q}=q$? Could someone explain to me the property?
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0answers
20 views

exponent laws for real numbers proof

I defined $a^b = \sup\{a^q:q\in\mathbb Q, q < b\}$ where $b$ is a real number and $a$ is a real number that is not smaller than $1$. When $ 0 < a < 1$, I defined $a^b$ as the reciprocal of $(...
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1answer
95 views

What is wrong with this argument that $(-2)^{1/4}=4^{1/8}$?

My prof started today's lecture by writing $$(-2)^\frac{1}{4} = (-2)^{(2*\frac{1}{8})} = ((-2)^{2})^{\frac{1}{8}} = 4^{\frac{1}{8}}$$ and asked us whether this was valid or not. However he didn't ...
0
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1answer
85 views

If $n$ is an rational, when is $n^{1/n}$ rational?

Given that $n$ is rational, when is $\sqrt[n]{n}$ rational? We can make a polynomial $x^{n}-n$ whose root is $\sqrt[n]{n}$ and using RRT we can show that there are no rational roots, but in the ...
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2answers
53 views

How do you solve $2^{x-1}=\frac{1}{x}$?

$2^{x-1}=\frac{1}{x}$ Clearly by substitute $x=1$ we were able to solve this problem but how do we really solve it using calculus?
3
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0answers
42 views

Solve the inequality $ a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc$

Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$ a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc\quad (1)$$ I have a proof : I was thinking for an alternative proof considering by example Young's ...
1
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1answer
73 views

Expansion of square root of a sum

I know that $(a + b)^2$ can be expanded as $(a + b) * (a + b) = a^2 + 2ab + b^2$. Is there an equivalent expansion method for the square root of a sum, that is, $(a + b)^{1/2}$? If there's no method,...
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1answer
46 views

Infinite Power Tower approximation: float error? [closed]

Desmos appears to plot it falsely using the $x^y = y$ definition, curving backwards. I've included a 50x exponent for comparison, which suggests no values flowing left in $x$-axis due to float error - ...
0
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1answer
59 views

Prove that for $a>2-2\log(2)$ the equation $e^x= 2x+a$ there are 2 distinct solution [closed]

I have to prove that $e^x= 2x+a$ for $a>2-2\log(2)$ there are 2 distinct solution. I've thought using Bolzano's theorem but I can't understand where to start, thanks in advance. Edit: the solution ...
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2answers
30 views

$e$ and $\ln$ : how to derive two equivalent equations

When solving the equation $$150 = 160 - 40 e^{-t/20}$$ I come to a solution that seems natural to me as follows: \begin{align*} .25 &= e^{-t/20}\\ \ln(.25) &= -t/20\\ t &= -20 \ln(.25)\\ &...
3
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3answers
63 views

Show that $3^{22}-2^{20}$ is divisible by $7$

I have this one question which after some hours of thinking I can't seem to be getting anywhere. The question reads: Show that $3^{22}-2^{20}$ is divisible by $7$. Now, after using a calculator I know ...
10
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3answers
177 views

Find all $x\in\mathbb{R}$ such that $\left( \sqrt{2-\sqrt{2} }\right)^x+\left( \sqrt{2+\sqrt{2} }\right)^x=2^x$.

Find all $x\in\mathbb{R}$ such that: $$ \left( \sqrt{2-\sqrt{2} }\right)^x+\left( \sqrt{2+\sqrt{2} }\right)^x=2^x\,. $$ Immediately we notice that $x=2$ satisfies the equation. Then we see that $LHS=...
1
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1answer
68 views

A difficulty in proving $Y^{T_1\oplus T_2}\cong Y^{T_1}\otimes Y^{T_2}$ in an arbitrary category

I've been struggling to write a complete proof following the guide provided in [1]. As showed in the book, I constructed the potential isomorphisms $\alpha:Y^{T_1 +T_2}\rightarrow Y^{T_1}\times Y^{T_2}...
5
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1answer
75 views

Verify my proof that for any $n>1$, if $n^n+1$ is prime, then $n=2^{2^k}$ for some integer $k$.

I am solving a problem and I am respectfully asking someone to critique my work and offer suggestions on formatting or point out any glaring logical errors. Here is the problem: Prove that for any $n&...
6
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1answer
57 views

Generalizing $a^ab^b>a^bb^a$, $a^ab^bc^c>a^bb^cc^a$

Title Generalizing $a^ab^b>a^bb^a$, $a^ab^bc^c>a^bb^cc^a$. Background We consider two distinct numbers $a,b\in\mathbb{R}^+$, no matter $a<b$ or $b<a$, we have: $$\frac{a^ab^b}{a^bb^a}=\Big(...
1
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1answer
47 views

Simultaneous multiple expansion

I am working on algorithm called as Simultaneous multiple exponentiation, I need to understand the mathematical meaning, like from $j=0$ to $k-1$, how, we calculate the value for $G_i$? What does the ...
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5answers
52 views

Exponential equation question (can't solve)

I came upon this question on a website: Find all the real solutions to $4^x-2^x=56$. I've tried to factor the expression: $2^x(2^x-1)=56$, but I don't know how to proceed. How can I solve this?
0
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1answer
26 views

Difference of compounding interest question

I am trying to see how compounding interest and inflation affects purchasing power. Let $x>y> n>1$. Does $x^n-y^n > (x-y+1)^n$ hold ?
0
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0answers
37 views

Why does the Gaussian function has an additional factor of 1/2 in the exponent?

What is the reason behind the seemingly arbitrary factor of $\displaystyle\frac{1}{{2}}$ in the exponent of the Gaussian function? Sometimes, as in this Wikipedia article on the Gaussian integral, the ...
2
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3answers
156 views

Is there a specific value of $0^0$? Is it undefined? [duplicate]

I was curious about the value of $0^0$ and I think it's either $1$ or $-1$. This is the reason: $$\text{Let } a=0^0.\\\frac{1}{a}=0^{-0} \\\text{where 0=-0}, a=\frac{1}{a}\\ a^2=1\\a=\pm1$$However, I'...
3
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2answers
62 views

The value of $\sum_{n=1}^{\infty}\frac{1}{n^n}$?

In my last question I asked about riemann zeta function $\zeta(3)$ and got an answer as apery's constant which is an unsimplifiable transcendental constant. I don't know why but I just got more ...
3
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2answers
70 views

Calculating the integral of exponential of exponential

I've been trying to integrate the following expression: $$\int_{0}^{\infty} (1- \exp(-a e^{-bx})) dx$$ I started to solve this integral by first using the substitution method. With $t = e^{-bx}$, then ...
6
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2answers
522 views

Show that : $f(x)+f(1-x)\leq 2$

I'm very proud to show one of my dream in term of inequalities . Claim Let $0.25\leq x\leq 0.75$ and $x\neq \frac{2k+1}{100}$ with $12\leq k\leq 37$ and $k$ a natural number then define the function :...
2
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4answers
136 views

If $a = \frac{1+\sqrt5}2$, then what is $a^{18} + \frac{323}{a^6}$?

My question is this: If $a = \frac{1+\sqrt5}2,\frac{1-\sqrt5}2$, then what is $a^{18} + \frac{323}{a^6}$? It is an AMC style question and is timed, so I will not be able to use solutions with a lot ...
2
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2answers
30 views

solve for $x$, $(\sqrt{a+ \sqrt{a^2-1}})^x+(\sqrt{a- \sqrt{a^2-1}})^x=2a$

Find the value of $x$ when $$(\sqrt{a+ \sqrt{a^2-1}})^x+(\sqrt{a- \sqrt{a^2-1}})^x=2a.$$See, by hit and trial method it is clear that $x=2$ is a solution. But I failed to solve this explicitly to get ...
0
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0answers
27 views

Maths notation: power raised after argument bracket

Apologies if this is a very basic question, but I came across a somewhat confusing notation and am not familiar with it. I am reading an article that contains the following notation in some of its ...
0
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1answer
54 views

Inequalities for generalized means

Given a set of $n$ observations $x_1,\cdots,x_n$, the power mean is defined as $$M_p = \Big(\frac{x_1^p+\cdots + x_n^p}{n}\Big)^{1/p}.$$ Likewise, the exponential mean is defined as $$m_p = \log_p\Big(...
0
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1answer
35 views

How to get the value of r when I have the value of r^² ( 1000 = r^2)?

I do not remember this from school anymore and I now I have to use the logarithm but the problem is that I do not know the base. I want to know the value of r but I only have the value of r^2. 2² = 2*...
2
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1answer
35 views

If there is a way to express an integer as a sum of three non-zero cubes, then there are infinitely many ways

How to prove the following statement: "If an integer can be expressed as a sum of three non-zero cubes in a way, then it can be expressed as a sum of three non-zero cubes in infinitely many ways.&...
0
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1answer
46 views

How to prove that $(a^n)^m = a^{nm}$ for non-integer exponents?

This is a question that has been twinging me for a while. I have been told that $\sqrt{a} = a^{1\over{2}}, a\in\mathbb{R_+}$ and ${1\over{a}} = a^{-a}, a\in\mathbb{R_*}$ were nothing more that ...
1
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0answers
76 views

Is there a known formula for the number of $k^{\text{th}}$ power residues modulo $2^n$?

We define a $k^{\text{th}}$ power residue modulo $n$ to be an integer $a$ coprime to $n$ such that there exists an integer $x$ satisfying $$x^k\equiv a\pmod{n}.$$ A fundamental question that we can ...
5
votes
1answer
89 views

Base-Exponent Invariants

A sum of powers is called a base-exponent invariant if its value does not change if each base and exponent are switched. The simplest example is $2^4$, which of course is equal to $4^2$. Another ...
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0answers
22 views

Series to power example

community. Found an example which seemingly easy but I cannot get the proper idea. Consider absolute convergent $\forall z$ series $\sum_{n=0}^{\infty}c_{n}z^{n}$ . Since $z^k$ function can be ...
0
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0answers
51 views

Does entropy increase exponentially as the number of observations increase?

Is there an example that demonstrates that Shannon's information entropy ($H(X)$ formula below) will always increase exponentially as the number of observations in the data increases (preferably using ...
1
vote
3answers
75 views

What is $\ln(-1)$? How is $\ln(x)$ defined on the negative complex numbers?

At first, this question - when I asked it for my self - for me was straight forward: $$e^{i\pi}=-1 \implies \ln(-1)=i\pi$$ Yet again, at some time I also discovered that: $$e^{i\pi}=-1 \iff -e^{i\pi}=...

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