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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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Why is $0.5^x + 0.5^x = 0.5^{x-1} $

I believe that the following is always true: $0.5^x + 0.5^x = 0.5^{x-1} $ but I do not know why. I've tried to prove it but am unsure how. Is it always true? And is there a proof?
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1answer
22 views

Find the least possible n for the power

While comparing the $123^{124}$ and $124^{123}$ (just for fun) I come up with an interesting question. Is it possible to find the least possible natural $n$, such that $a = b - k, k > 0 \quad and \...
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1answer
31 views

multiplication with exponentials [on hold]

How is this correct? $$\hbar\omega e^{iHt/\hbar}\left(a_S^{\dagger}a_S+\frac{1}{2}\right)e^{-iHt/\hbar}=\hbar\omega\left(e^{iHt/\hbar}a_S^{\dagger}a_Se^{-iHt/\hbar}+\frac{1}{2}\right)$$ Source: http:...
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2answers
33 views

Method for finding a sequence's limit

i'm trying to find the following limit, if it exists, $$\lim_{n→ ∞} \frac{(n+7)^{n-5}}{n^n}$$. Now, I've tried division like $$\lim_{n→ ∞}\frac{|n+1|}{|n|}$$, or dividing by the highest power, but I ...
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1answer
44 views

How do I isolate $k$ in the following equation?

I tried to isolate the unknown $k$ in the the following equation by using logarithms, but my resolution was ugly. I am trying to learn mathematics by my own (again!), because it's a beautiful subject ...
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2answers
78 views

Nice power sum inequality $ab(a-b)\leq a^{ab}-b^{ab}$ for $a+b=2$

Let $a\geq b>0$ such that $a+b=2$ then we have : $$ab(a-b)\leq a^{ab}-b^{ab}$$ My try : We study the function $f(x)$ on $[0;1]$ such that : $$f(x)=(2-x)^{((2-x)x)}-x^{((2-x)(x))}-(2-x)x(2-2x)$$ ...
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1answer
26 views

Matrix for recurrence relation

I have recurrence relation for which i need to figure out matrix which can used for matrix exponentiation. $f(n) = f(n-1) + f(n-3)$ Which I think can be written as $f(n) = a f(n-1) + b f(n-2) + c f(...
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3answers
60 views

If $2^{k+1} = 2 \cdot 2^k$ what does $2^{k-1}$ equal?

I know that $2^{k+1} = 2 \cdot 2^k$, but what does $2^{k-1}$ equal? Is it $\frac{2^k}{2}$? Then does $2^{k-2} = \frac{2^k}{2^2}$?
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2answers
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Find $A^5-4A^4+7A^3+11A^2-A-10I$ [duplicate]

If$$A= \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix},$$then find $A^5-4A^4+7A^3+11A^2-A-10I$ where $I$ is Identity matrix of $2^{nd}$ order. Answer should come in terms of $A$ ...
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Properties of logarithms and exponents

I had a question about the properties of logarithms and exponents, that I need some assistance on: If you are familiar with electrical engineering principles, you may be familiar with Shannon's ...
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0answers
28 views

Raising consecutive numbers to the power of 2 (or 3) [duplicate]

While I was falling asleep last night, I thought of an interesting mathematical pattern (which I hope is not something trivial that they teach you in school). In short, an example: $1^2 = 1$ $2^2 = ...
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5answers
976 views

Do powers of 256 all end by 6 and if so, how to prove it? [duplicate]

I computed the 10 first powers of 256 and I noticed that they all end by 6. ...
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1answer
52 views

Sum of Sine and Cosine to Higher Powers Is Constant (Recursion, Proofs)

I have observed this empirically, but I have no idea how to prove it or if it has been proven before. If this has been proven before, in any form, either more or less generic, please point me to such ...
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2answers
31 views

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $ (The second sum in the left-hand side of the inequality is an exponent) I've ...
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Modular Exponentiation Speedup

Is it possible to speedup modular exponentiation $a^e\bmod n$ with large $n$, $e$ when the complete factorization of $n-1$ is known?
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1answer
41 views

Finding limit when values of derivatives at a point are given

Let $f:\Bbb R\to\Bbb R$ be such that $f''$ is continuous on $\Bbb R$ and $f(0)=1, f'(0)=0, f''(0)=-1$. The $\displaystyle{\lim_{x\to\infty}\left(f\left(\frac{\sqrt2}{x}\right)\right)^x}$ is ..... I ...
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2answers
46 views

Addition with variable in exponent

I know that $2^x$+$2^x$=$2^{x+1}$ but I can not explain why. Googling returns too much information about the situation involving variable in the base. Can someone explain? A link to a reference ...
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1answer
65 views

Is there any better simplified form of $a^m-(a-1)^m$ [closed]

I am studying some algebraic identities and i wonder if there is any simple form of above equation. $a,m$ are positive integers,
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0answers
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Is there any simple ways to compare $x^y$ and $y^x$ without a calculator?

There are plenty of discussion on MSE about how to compare $x^y$ and $y^x$. For $x,y>e$, it is sufficient to just compare $x$ and $y$ to reach a conclusion. But I wonder if there are some general ...
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5answers
233 views

How do you prove that integer powers of $1.5$ are not equal to any integer powers of $2$?

I've read somewhere that no integer power of $1.5$ can ever equal any integer power of $2$ (besides the zeroth power, of course). It makes sense, but how is this proven? (The application here is ...
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3answers
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Complex operator $i$ and Exponents

I am trying to understand the complex numbers and exponents. I came across this question. I wonder how to explain the difference between $${2\cdot i} \text{ and } 2^i$$ as $i=\sqrt{-1}$ edit: ...
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2answers
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Why does $(-n)^{4.4}$ yield a non-real number in WolframAlpha? $n$ stands for any number

One would think that, since $4.4$ as a mixed fraction is ($22/5$), raising a number regardless of it being negative or positive would yield a real number. However, when plugging this equation into ...
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0answers
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Closed forms of binomial sums and power identities

Proposition 1.1 For every integers $m,n\geq 0$ the following identity holds \begin{equation} n^{2m+1}=\sum_{k=1}^{n}\sum_{j=0}^m A_{m,j}k\strut^j(n-k)\strut^j=\sum_{k=0}^{m}(-1)^{m-k}U_m(n, k)\cdot n\...
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1answer
36 views

Weaknesses in determining power law properties from data using weighted average slope?

I'm attempting to determine the best approximation for the a model for a data set with power law characteristics as illustrated below (blue scatter). The model (the orange line) has been determined ...
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1answer
17 views

How do I calculate an x with a circle in it and a variable raised to -T?

I am trying to design a decoupler to break a MIMO system down to a distributed SISO system for control. I am reading a paper on how to do that, but it is telling me to compute some strange stuff that ...
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1answer
40 views

Proving $b^a(1-b)^{1-a} \leq a^a(1-a)^{1-a}$ where $0 \leq b \leq a \leq 1$

I am trying to prove an inequality which boils down to the following: $$b^a(1-b)^{1-a} \leq a^a(1-a)^{1-a}$$ where $0 \leq b \leq a \leq 1$ Another equivalent formulation is: $$ a\log{b} + (1-a)\log{(...
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1answer
24 views

What does it mean when the exponent is beneath the base number?

I am familiar with the exponential function and the power function. Exponential Function: $y=3^x$ Power of Function: $y=x^3$ But what do the numbers under the theta in this image represent? $h_𝜃(...
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1answer
22 views

What is the power of a modular exponent expression? [closed]

Is there any property to simplify expressions of the following type: $(a^{x}\ mod\ N)^y$ The usual mathematical property $(a^{b})^c = a^{bc}$ doesn't seem to hold.
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19 views

How to remove power in a modular exponent expression?

I am implementing a cryptography scheme which involves verifiying some data through the following process: Suppose party A wants to verify data held by party B Party A has: $a^x mod\ N$ Party B has:...
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1answer
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Why is $-1=e^{\pi i }=e^{2\pi i\frac{1}{2}}\neq (e^{2\pi i})^{\frac{1}{2}}=1^{\frac{1}{2}}=1$ true?

I thought there was this rule that $e^{xB}=(e^x)^B$? Also what I don't understand is that $x^{\frac{1}{2}}$ is defined as the square root of $x$. And because the square root may also be in $\mathbb{C}...
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1answer
48 views

Different results with different methods to same limit of polynomial

I'm considering the following limit while looking for asymptotes: $$\lim_{x\to \infty} \arctan\dfrac{\sqrt{x^2+1}}{x-1}$$ Going the route I initially tried, I divide both parts of the fraction by $x^...
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5answers
71 views

How to simplify $2^{(3+(4+(5+(6+(7+(8))))))}$? [closed]

Thanks to the rules of parenthesis in exponents, $2^{(3+(4+(5+(6+(7+(8))))))}$ has a huge number of parenthesis. How do you simplify this so you wouldn't have to write $))))))$ at the end of a math ...
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3answers
58 views

Prove that when $a,b \ge 0, r \ge 1$ that $(a + b)^r \ge a^r + b^r$

Prove that when $a,b,r \in \mathbb{R}, a,b \ge 0, r \ge 1$ that $(a + b)^r \ge a^r + b^r$ My first idea for this proof was to use the generalized binomial theorem: \begin{align*} (a+b)^r &= \...
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0answers
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Complex Roots question.

How many complex solutions does 2^π have? Obviously, something like 2^(7/3) has 3 complex solutions, ie. 3 complex numbers c such that 2∈ c^(3/7). What is the set of c such that 2∈c^(1/π)? Is it ...
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1answer
16 views

Simplification with exponents

I'm currently revising an exam about channel coding in telecommunications and we have a question where we need to isolate a variable $u$ in terms of another variable $q$. Currently, I am stuck with a ...
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4answers
64 views

If $(1 + x)^{10} = 2.5$, what is $x$?

If $$(1 + x)^{10} = 2.5$$ , the next step would be $$10 * \log(1 + x) = \log(2.5)$$ $$\log(1 + x) = \frac{\log(2.5)}{10}$$ . This is where I am stuck. I know that $$\log_{b}(a + c) = \log_{b}(a) ...
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2answers
43 views

Smooth transition between functions

Therefor that I don't really have a mathematical background, it is kind of difficult to me, to describe what I'm looking for (but I'll give it a try): I'm looking for a way to parameterize a function ...
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1answer
28 views

Modulo of a power of a power of a power

I have a question related to a modular exponentiation. My question is this: let a equal $8*12^\frac{{3^{9998}-3}}{2}$. Let b equal $8*12^\frac{{3^{a-2}-3}}{2}$. Finally, let c equal $8*12^\frac{{3^{b-...
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1answer
69 views

How to parse $2^{1+2^3}$?

I'm fairly comfortable with arithmetic and high school math overall, but I'm having trouble being 100% sure that $2^{\large {1+2^3}}$ should be parsed as 2^(1+(2^3))...
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1answer
140 views

How to add numbers with same base but unknown exponents?

I'm taking a university math course despite knowing better. Among the straightforward problems I encountered the following: Simplify: $$5^x +5^{x+2}$$ The answer is supposed to be $26*5^x$ How am ...
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0answers
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Why is rational exponentiation an algebraic operation?

I can't manage to understand what is the main criterion for an operation to be algebraic. I orginally thought it was an operation that could be expressed via the standard arithmetic operations (...
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0answers
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What's the power of quaternion $[e^{a} * (cos(r) + \frac{sin(r)}{r}(bi+cj+dk)) ]^ 2$?

Euler's identity extended into quaternions is: $q = a + bi + cj + dk$ with a,b,c,d real numbers for the below: $\sqrt{b^2+ c^2 + d^2} = r > 0$, and $\frac{bi+cj+dk}{r}$ = $\sqrt{-1}$, ...
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0answers
27 views

Exponent operations: addition, multiplication, grouping etc.

I get easily confused while evaluating exponents: when should I add them and multiply them? We add exponents when multiplying two numbers are of the same base: i.e., $ (2^2) . (2^3) = 2^5 $ - When ...
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LTE applications

Let k be a positive integer. Find all positive integers n such that $3^k | 2^n - 1$ I just started reading and learning about the Lifting the Exponents Lemma, and I want to try and use it in this ...
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1answer
43 views

Subtracting exponents properties?

Background : I was reviewing some practice problems for a small local math competition, and I don't understand how the give solution works. I don't know how this, $$9^{x+2} - 9^x=240$$ is the same ...
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2answers
114 views

Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational.

In solving the following problem: Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational. I let $x=2$ and $y = \sqrt 2$, so that $x^y = 2^\...
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1answer
24 views

How was 'DF' calculated? [closed]

Asking formulae to how to calculate variable DF ..I would like to know how 'DF' was calculated as below my screenshot: enter image description here unknow: DF, DFCF Known: # of Year, Rate=10%, FCF, ...
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1answer
51 views

Show whether or not the functions $f(x,y)=|x|^y$, $g(x,y)=|x|^{|1/y|}$ have limits at $(0,0)$.

Show whether or not the functions $f(x,y)=|x|^y$, $g(x,y)=|x|^{|1/y|}$ have limits at $(0,0)$. By the answers only the latter has a limit at $(0,0)$ but I don't know how to prove.
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1answer
36 views

Connection between powers of $i$ and derivatives of $\sin$ and $\cos$?

There are plenty of connections between trigonometry and the complex plane. One particular comparison seems striking to me, and that’s the four-step cycle inherent in exponentiating $i$ to various ...
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1answer
20 views

How to write $12 (N / m)^2$ in long form

I want to write the value $12 (N/m)^2$ in long form like "12 Newtons per meter squared". However, I believe that because of the order of operations, my long form value will be interpreted as $12 N/(m^...