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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Log power rule problem

According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian plane....
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How can I simplify $(n\,\mathrm{ mod }\,b^{k+1})\,\mathrm{ mod }\, b^k$? [duplicate]

I am trying to simplify an expression I've found that is related to converting from a number base to another: $$n\,\mathrm{ mod }\,b^{k+1} - (n\,\mathrm{ mod }\,b^{k+1})\,\mathrm{ mod }\, b^k$$ In ...
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Does it seem right to calculate fractional powers by walking steps method?

We know that $x^y$ means $x$ multiplied by itself $y$ times or $x$ taking $y$ steps. Is this method suitable when $y$ (the power) is fractional? e.g. if $x=2$ and $y = 3.5$, we would have $$2^{3.5} = (...
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1answer
39 views

Using the number e to find the limit of a sequence [closed]

I have been trying to solve this problem for the past day and I just can't get my head around it, I know I need to use Euler's formula but how do I proceed after simplifying it. $$ B_n = \left( \frac{...
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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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1answer
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Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f : \mathbb R \to \mathbb R \text , $$ $$ f \left( a ^ b \right) = f ( a) ^ { f ( b ) } \text , $$ when $ a , b \in \mathbb R $, $ a , b \ge 0 $. So far the ...
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1answer
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How to do exponential division when base and exponent is different between numerator and denominator?

How to do division when (a power x) / (b power y) Say, a,x,b,y are distinct values. For example, (2 power 19) / (7 power 23). Doing mathematical operations with bigger exponents is tedious and ...
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2answers
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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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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.
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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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Prove that an inequality holds for $\alpha \ge 2$

It's related to my own question Refinement inequality of : $\sqrt{x}+x^{\frac{x}{x+1}}\geq x+1$ Let $x\geq 5$ be a real number then we have : $$\frac{x^2+1}{x+1}\Bigg(\frac{x^{\frac{x}{x+1}}}{x^{\...
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1answer
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Radioactive Decay formula is $A=A_0e^{-kt}$. How many years until 10 grams decay so that only 8 remain

I have been trying this question for hours and come to a dead end every time... Consider the radioactive decay formula $A=A_0e^{-kt}$ where $A$ is the amount of radium remaining at the time $t$. $A_0$...
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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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1answer
175 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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2answers
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How to prove $x^ax^b = x^{a+b}$

I am looking for a proof of one of the exponent combination laws, namely the sum of powers. Here $x, a, b \in \mathbb R$ and $x > 0$. I thought about induction but since a,b are not only positive ...
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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
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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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7answers
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Finding $\int x^xdx$

I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$. $$\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] &=\dfrac{\left(x^...
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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
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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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Why don't we use subscript for roots?

I was wondering, why do we use a radical sign $\sqrt{}$ for roots? Why don't we use subscripts instead? It would make sense too, regarding the fact that $\sqrt{5^3} = 5^{3/2}$. A switch like that ...
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Prove that $a^x$ is continuous

I'm having trouble with proving the following: Let $a > 0$ be a positive real number. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) := a^x$ is continuous. I'm a ...
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1answer
34 views

'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
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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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0answers
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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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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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Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also,...
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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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2answers
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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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0answers
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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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The binomial formula and the value of $0^0$

Here is the text from Knuth's The Art of computer programming, 1.2.6 F formula 14: Knuth doesn't give the proof of the statement. So, I tried to write it myself. To make binomial formula equal to $...
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1answer
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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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1answer
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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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1answer
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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
48 views

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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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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2answers
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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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1answer
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Why does this exponential identity hold? $3\cdot 2^{2n+2} - 3\cdot2^{2n} = 3\cdot2^{2n}$

Can someone explain how this is valid? $$3\cdot 2^{2n+2} - 3\cdot2^{2n} = 3\cdot2^{2n}$$ Thanks so much!
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1answer
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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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1answer
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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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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
66 views

Complete the square for exponents

How does $\exp(2x)-2+\exp(-2x) = (\exp(x) - \exp(-x))^2$ ? I am having trouble using complete the square.
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2answers
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Limit of $x^x$ as $x$ tends to $0$

I am trying to solve the following limit: $$\lim \limits_{x\to0} x^x$$ The only thing that comes to mind is to write $x^x$ as $e^{x\ln{x}}$ and getting the right sided limit would be easy but I don'...
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1answer
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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
115 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 \...
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1answer
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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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