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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13 views

Mean/median/etc of a zero-mean random variable with exponent

Suppose $\epsilon \sim N(0,\sigma^2)$, $a\in(0,1)$ and $x>0$ are constant. Is there any way of estimaing the following: $$(\frac{x+\epsilon}{x})^a$$ Although $\mathbb{E}[\frac{x+\epsilon}{x}]=1$, I ...
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47 views

Definition of $x^{-a}$ and $x^{1/a}$

So, the other night I was wondering why $x^{-a}=\dfrac{1}{x^a}$ and $x^{1/a}= \sqrt[a]{x}$, because in school the teacher (at least for me), gave this formulas without any explanation whatsoever. This ...
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I need to prove that $(a^n)(b^n) = (ab)^n,$ where $a,b\in\mathbb N$. Proof by induction on $n \in \mathbb Z, n\geq 0$.

I need to prove that $(a^n)(b^n) = (ab)^n,$ where $a,b\in\mathbb N$. Since $a^0=1$, is given in the question, I assumed my base step to be when $(n=0)$. $(a^0)(b^0)= (1)(1) =1.$ and now I'm stuck. I ...
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1answer
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Evaluating difficult integral with exponentials

I'm trying to evaluate a difficult integral. I'm able to break it down in separate terms and deal with scalar multiplication. However, I'm stuck trying to evaluate two terms in particular. Here is the ...
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3answers
40 views

HowTo Rearrange The Piano Key Frequencies Formula

The Piano Frequencies formula maps the position of a piano key on the standard piano n to a specific note and frequency e.g. key ...
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2answers
19 views

Exponentiation on a proof by induction

I am beginning to learn about proofs by induction, but cant get my head around the steps taken in a slide we've been shown. My problem lies within the red box, and I'm not sure if I'm missing some ...
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Why is $(e^{2πi})^i$ different from $(e^i)^{2πi}$ [duplicate]

I'm a high school senior and this question has been on my mind since 8th grade, since I've learned Euler's formula but I've thought about it a lot more recently. According to Wolfram Alpha: (e^i)^(2iπ)...
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1answer
171 views

Partial proof by induction of the inequality : $(1-x)^{(2x)^n}+x^{(2(1-x))^n}\leq 1$

Claim Let $0.5\leq x<1$ and $n\geq 2$ a natural number then we have : $$(1-x)^{(2x)^n}+x^{(2(1-x))^n}\leq 1\quad (I)$$ Sketch\Partial (of) proof . We use a form of the Young's inequality or ...
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1answer
33 views

Can you take a fractional root of a number?

Is it possible to say, take a 1,5th root of 2? And if not why so? Wouldn’t a 0,5th root of a to the third power = a to the power of 3/0,5 = a to the power of 6? If this is not allowed than why?
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For which positive integers $x$, $y$ satisfy the following equation: $x^2 + y^2 = 2020$?

This problem is driving me absolutely crazy. I managed to determine the max value of $x$ and $y$: $x^2 + y^2 = 2020$ $=>x^2 = 2020 - y^2$ It's obvious that square cannot be smaller than 0, and we'...
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1answer
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'Vandermonde-esque' matrix has nonzero determinant.

I want to show that the matrix $$M=\begin{pmatrix} z_1^{y_1} & z_1^{y_2} & \cdots & z_1^{y_n} \\ z_2^{y_1} & z_2^{y_2} & \cdots & z_2^{y_n} \\ \vdots & \vdots & \...
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Refinements of the inequality $f(x)=x^{2(1-x)}+(1-x)^{2x}\leq 1$ for $0<x<0.5$

Hi it's related to Showing the inequality $f(x)=x^{2(1-x)}+(1-x)^{2x}\leq 1$ for $0<x<1$ We want to show 1: Let $0<x<0.5$ such that then we have : $$f(x)=x^{2(1-x)}+(1-x)^{2x}\leq q(x)=(...
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Restrictions on exponential laws

Wanna be sure this is fully correct: So i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the laws i want to write ...
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1answer
41 views

Restrictions on exponential

This question has already been asked but no one answered so: so i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the ...
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Is it the case that there is some string, S, such that every string ending in S ends a perfect square? If so, is the same true for every power?

I am not a mathematician, but I gather from this paper that any string of digits ending in '1', '3', '7', or '9', no matter what it is, is the ending sequence of the decimal representation of ...
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1answer
27 views

Rewrite $ \frac{2}{5}(x-2)^\frac{5}{2} + \frac{4}{3}(x-2)^\frac{3}{2} + c$ to $\frac{2}{15}(3x+4)(x-2)^\frac{3}{2} + c $

I must get the expression $ \frac{2}{5}(x-2)^\frac{5}{2} + \frac{4}{3}(x-2)^\frac{3}{2} + c$ into $$ \frac{2}{15}(3x+4)(x-2)^\frac{3}{2} + c $$ I tried expanding the $\frac{4}{3}(x-2)^\frac{3}{2}$ ...
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Exponents and Log properties

This is part of a proof I am working on. So on one line, I needed to show that $$2^{\Bigg(\cfrac{\ln\frac{1}{n}}{\ln(2)}\Bigg)} =\frac{1}{n}$$ That was $2$ to the power of $\ \ \cfrac{\ln\frac{1}{...
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Restrictions adding exponent laws

Alright, so i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the laws i want to write restrictions on. SO, please let ...
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3answers
173 views

Is there a solution to $2^a+2^b = 10^c+10^d$, with $0 \leq a < b$ and $0 \leq c < d$?

This question arose on the code golf StackExchange: Is there a solution to $2^a+2^b = 10^c+10^d$, with $0 \leq a < b$ and $0 \leq c < d$? In other terms: is there an integer that looks like $\...
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Significance of $e^\pi$

I know that $$\sqrt{e^\pi}=\sqrt[i]{i}$$ which is quite a nice result, and that it is transcendental. Does the constant $e^\pi$ have any other significance? Thanks. I realize this is quite a trivial ...
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The number of digits in an exponent of two

At the moment my area of research is Mersenne Primes. I am currently looking into a specifically large exponent, $\ 2^{112401533}-1$, and I can’t help but wonder how I would calculate how many digits ...
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What is $-1$ to the power of a fraction?

If I have any negative number (does not have to be $-1$), how would I determine that value to the power of $1/2$? I learned that $x^\frac{1}{2}$ is equal to $√x,$ so wouldn't $-1^\frac{1}{2}$ be equal ...
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1answer
34 views

Restrictions on exponent laws

Alright, so i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the laws i want to write restrictions on. SO, please let ...
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2answers
22 views

Understanding the evaluation of a fraction

Sorry if this is a little simple to ask here; but I'm at a loss as to where else to look for help with this one. I've been attempting to understand the evaluation of a fraction. The question I'm ...
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2answers
58 views

How to solve for $x$ in the equation $y = (x + 25)^x$ where you know the value of $y$

I am trying to figure out how to solve for $x$ in the equation $y = (x + 25)^x$ for any known $y$. For example, I know that in the example $2758547353515625 = (x + 25)^x$, the value of $x$ is 10. Is ...
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1answer
100 views

Reasoning about $2^m - 3^n$ where $m>0,n>0$ are integers

Let: $m>0, n>0, c>0$ be integers Does it follow that: $$2^m - 3^n = \begin{cases} -3^n(1 - 2^{-c}) + \sum\limits_{i=0}^{n-1} \left(3^{n-1-i}2^{2i-c}\right), & m=2n-c\\ \sum\limits_{i=0}^...
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Exponents and bases

I have some doubts in understanding the necessity of some steps in the following situation. While solving an algebraic problem, I came across an expression of the following type, where I have to solve ...
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1answer
208 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 case ...
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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 \...
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1answer
16 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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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} = -...
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1answer
28 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 ...
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1answer
33 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 ...
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28 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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1answer
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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 ...
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88 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 ...
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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 ...
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1answer
28 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
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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 ...
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2answers
61 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} \...
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1answer
59 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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0answers
30 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
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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 ...
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1answer
87 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
56 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?
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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 ...
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1answer
93 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 - ...
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1answer
60 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)\\ &...

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