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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42 views

Ambiguity in product of all $n$th roots of unity

I was studying the product of $n$th roots of unity, and the proof in the book went like this: $$1\cdot a\cdot a^2\cdot\cdots\cdot a^{n-1} \\=a^{0+1+2+3...+n-1} \\=a^{n(n-1)/2} \\=(\cos\frac{2\pi}n+i\...
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3answers
94 views

Integer solution for $2^{3p}+2^{4q}=2^{5r}$

If $x,y,z$ are integer powers of $2$, prove that equation $$x^3+y^4=z^5$$ does not have solution, otherwise give a counterexample. I tried to do this exercise, I firmly believe that the equation is ...
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4answers
141 views

If $x^{x^{x+1}}=\sqrt{2}$, then evaluate $x^{x^{p}}$, where $p = 2x^{x+1}+x+1$

I can't figure out how to give a proper form to this expression to use the root of two. If $$x^{x^{x+1}}=\sqrt{2}$$ find the value of $W$ if $$W=x^{x^{p}} \quad\text{where}\; p = 2x^{x+1}+x+1$$ EDIT:...
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51 views

What is the name of this method?

I have been doing some research into discrete log problem in the form of $$f(x) = g^{\operatorname{floor}(x)} \; \text{mod } p.$$ Interestingly I found that there is a function $t(x)$ where $t(f(x))= ...
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1answer
38 views

If $D$ is diagonal matrix, then why $e^{D}$ turns out to be like this…

If $D$ is a diagonal matrix, for instance $D=\begin{pmatrix}a&0\\0&b\end{pmatrix}$. Im wondering why $e^{D}=\begin{pmatrix} e^{a} &0\\0& e^{b} \end{pmatrix}$. I already know that $e^{x}...
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2answers
28 views

Finding the $i$th power of $i$ [duplicate]

I have been trying to find the $i$th power of $i$ as a mental exercise, I have tried two approaches For the first, using properties of exponents $$i^i=i^{\sqrt{-1}}=i^{-1^{\frac{1}{2}}} \implies i^i=i^...
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1answer
44 views

Why does zero raised to any positive power equal zero? [closed]

What if you raised 0 to the power of 2?
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2answers
105 views

How to prove solutions of $2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2$?

The original question wanted the real solutions to $$ 2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 . $$ I graphed this equation and worked out the trivial roots $x=-1$ and $x=1$, and I could not find any ...
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1answer
30 views

Why is $\exp(n/2 * (\log 2)^2) \approx 1.27^n$

I saw this approximation made in https://stats.stackexchange.com/questions/473496/infinite-coin-toss-probability by the accepted answer. What inspired this approximation?
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44 views

Why is $f(x)$ undefined for negative $x$ on Maple and my calculator but defined on WolframAlpha?

$f(x):=x^{4/5}\cdot (x-4)^2$ Maple gives me the graph: WolframAlpha gives this graph and my calculator gives Error 2 if I plug in $(-1)^{4/5}$. My original intuition would be to think that the graph ...
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1answer
54 views

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$. My try It is easy to see that if we raise the first equation ...
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1answer
18 views

What is the proper convention regarding the order of operations of a fractional exponent and/or the simplification of it?

Specifically, consider the example $\sqrt[4]{x^2}$. The answer of course would be $\sqrt{|x|}$ since the x is squared first. However if converted to the exponential fraction of $x^{2/4}$, you lose the ...
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0answers
29 views

$\lim f(t)^{g(t)} = 0^0 = 1$ if $f,g$ are analytic?

Wikipedia states that if $f,g$ are real analytic near $c$, and $f,g \to 0$ as $t \to c$, then along any path for which $f>0$ one has $f(t)^{g(t)} \to 1$. The proof Wikipedia cites is in German so I ...
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2answers
43 views

Converting a radical to a fractional exponent

I want to understand how to convert a radical to a fractional exponent. Given the following equation: $\sqrt[3]{(x)^6\cdot x^9}=\sqrt[3]{x^{24}\cdot x^9}=\sqrt[3]{x^{33}}=x^{\frac{33}3}=x^{11}$ How ...
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0answers
13 views

Upper bound on sum of exponential in a quotient in a matrix form

Given $K > 0$, $x \in \mathbb{R}^d$, $A \in \mathbb{R}^{d \times m}$, where $a_i^T \in \mathbb{R}^d, i = \{1,.., m\}$ is row vector of A. $$ f_{j,k}(x) = \frac{\sum_{i=1}^{m} a_{ij} a_{ik} e^{K(...
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1answer
221 views

Arc Length Integral of $x^x$ from 0 to 1 in closed form.

I was recently trying to compute the arc length of $x^x$ from $0$ to $1$ as follows: $$L=\int_0^1 \sqrt{1+\left(\frac{\text{d}}{\text{d}x}x^x\right)^2} \text{d}x=$$ $$\int_0^1\sqrt{1+x^{2x}(\ln x+1)^2}...
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0answers
35 views

Find all nonnegative integer solutions to $2^a +3^b +5^c =n!$

Question: Find all nonnegative integer solutions to $2^a +3^b +5^c =n!$. My work so far: I realize that if $n$ is greater than $2, 3$, or $5$, then $3^b+5^c$ has to be divisible by $2$, $2^a+5^c$ has ...
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1answer
64 views

Compound interest - relationship between $\frac{r}{n}$ and $r$?

The compound interest formula $A=P\left(1+\frac{r}{n}\right)^{nt}$ is usually used in examples where you are given a nominal annual rate and then calculate the accrued amount, where $\frac{r}{n}$ is ...
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1answer
23 views

What is a more rigorous definition for powers of rational numbers?

My current understanding of $x^\frac{m}{n}$ is that it is equal to $\sqrt[n]{x^m}$. Now technically, (-1)$^\frac{2}{4}$ is equal to $\sqrt[4]{(-1)^2}$=1. As $\frac{2}{4}$=$\frac{1}{2}$, (-1)$^\frac{1}{...
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2answers
36 views

Finding powers in a power of two equation. [closed]

Find all $n$ such that there exists $x,b$ such that $$2^{n!} + 2^{n!-1}+\cdots+1 -x = 2^x + 2^{x/b}$$ where $b \mid x.$ I have considered units digits, as the RHS is $2^{n!+1}-1$ but I can't get any ...
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4answers
61 views

Solve the equation $11^{2(\log_5(x))^2} - 12 \cdot 11^{(\log_5(x))^2} + 11 = 0$ [closed]

Solve the equation $11^{2(\log_5(x))^2} - 12 \cdot 11^{(\log_5(x))^2} + 11 = 0$. I do not know what can I do with the ${(\log_5(x))^2}$ part. Could you provide a hint?
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1answer
28 views

Is there a proof that $x^{y+k} > (y+k)^x$ for $k>0$ with $x<y$ and $x^y > y^x$

I've come across this observation when I was trying to solve a programming task and I was wondering if there's a proof for it. I tried proving it myself but lacking a proper mathematical education I ...
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3answers
55 views

Pattern for $x^n + \frac{1}{x^n}$ if $x + \frac{1}{x} = \frac{1+\sqrt 5}{2}$ [duplicate]

I was trying to solve this problem on my textboox Given $x + \frac{1}{x}$ = $\frac{1+\sqrt{5}}{2}$ find the value of $x^{2000} + \frac{1}{x^{2000}}$ After doing a bit of exploration i have noticed ...
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0answers
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Calculate $2^{1000000} \mod 7$ [duplicate]

I don't know how to do this one. I found a similar problem here (#5), but I do not understand the given solution in the slightest (apart from calculating $\varphi (77)$). Another similar problem I ...
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1answer
61 views

How to estimate fractional exponents using only addition, subtraction, multiplication and division?

My question is related to Calculating logs and fractional exponents by hand, but despite that question's title, all answers there focus on logarithms. I am interested in fractional exponents alone. ...
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1answer
20 views

Question regarding proof of: For any x and any base b, $x = log_{b}(b^{x}$)

I am reading 'A beginners guide to discrete mathematics'. And i stumbled on this proof they have: Theorem. For any x and any base b, $x = log_{b}(b^{x})$ Proof. By definition, $u = b\text{ }log_{b}(u)$...
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1answer
35 views

Squaring a Gamma Function

Can anyone tell me if there are any implications of squaring a gamma function as shown below? $$\left(\frac{L\Gamma(8)\Gamma(\frac{11}{2})}{\Gamma(\frac{9}{2})\Gamma(9)}\right)^2$$ I just want to know ...
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1answer
44 views

Explanation for: $\frac{u^{-2}}{v^{-3}} = (u^{-2})(v^{-3})^{-1} = u^{-2}v^{3}=\frac{v^{3}}{u^{-2}}$ needed

I am just working through some sample problems in a introduction to discrete mathematics. And i encountered this problem: Task: Express in the simplest possible form, with positive exponents Concrete ...
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1answer
21 views

Gradient of a scalar to a power

I was working on a problem and had a question about how gradients worked. I have the following expression:$\nabla (\rho^{-\gamma})$ and was wondering if I could express it as: \begin{gather} \...
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0answers
23 views

Numerically stable computation of logarithm of sum and fraction of exponentials

I am trying to compute the following expression numerically stable: $$ \log \left(\frac{5 e^{a+b}+3 e^a+3 e^b-3}{-e^{a+b}+e^a+e^b+7}\right) \text{ for }a,b\in\mathbb{R} $$ I have attempted simplifying ...
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1answer
78 views

Solving $(\sqrt{2})^x+(\sqrt{2})^{x-1}=2(2\sqrt{2}+1)$

I'm in stuck with this simple equation. $$(\sqrt{2})^x+(\sqrt{2})^{x-1}=2(2\sqrt{2}+1)$$ This is my solution: $$\begin{align}(\sqrt{2})^x+(\sqrt{2})^x(\sqrt{2})^{-1} &=4\sqrt{2}+2 \tag{1}\\[4pt] 2^...
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0answers
57 views

How can I prove that this equality is not valid for any natural number?

this is an old competition problem I am struggling with. How can I prove that the equality $2^m = 4096c^3 + 192c^2+3c+1$ is not valid for any natural numbers c and m(m can be 0). This equation has a ...
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1answer
116 views

How to factor a modulus into 2 primes given 2 equations with the factors as the 2 unknowns?

Given the following $N = p*q$ and the following 2 equations $r1 = (a*p + b*q)^x \mod N$ $r2 = (c*p + d*q)^y \mod N$ $r1$, $r2$, $a$, $b$, $c$, $d$, $x$, $y$ and $N$ are all known. The only unknowns ...
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2answers
57 views

For which $n$, will $n^x$ and $x$ be equal for just $1$ value of $x$

So, I was playing around in desmos graphing calculator with functions of type $n^x$. I began by plotting $y=2^x$ and $y=x$. I saw that $2^x$ never equals x(kinda obvious but I am still a high schooler)...
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3answers
62 views

Solving $5\times9^x-8\times15^x+3\times25^x>0$

Question: Solve the inequality: $5\times9^x-8\times15^x+3\times25^x>0$ So far I have managed to factorise the inequality: $(5\times3^x-3\times5^x)(3^x-5^x)>0$ From here on, I am stuck. I would ...
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0answers
84 views

How to solve $x^x = x?$ [duplicate]

Solve: $x^x = x\quad (*)$ I can only solve it when $x > 0$ $(*)\Leftrightarrow \ln(x^x) = \ln x$ $\Leftrightarrow x\ln x - \ln x = 0$ $\Leftrightarrow \ln x(x -1) = 0$ Then $x = 1$ How can I solve ...
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1answer
62 views

Solving $A = 94\left(1 - e^{-0.43 B}\right)^{0.52}$ for $B$

I am not a regular math user. I need to solve an equation to apply on timeseries variables. I have value of $A$ but don't have value of $B$ variable. Here, time series variables (mentioned as variable ...
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3answers
136 views

Prove $(AB)^{k} = A^{k}B^{k}$

Apologies if this question was posed already. I need to prove that $(AB)^{k} = A^{k}B^{k}$ holds if $AB=BA$. After trying it myself, I looked at the solution and found it quite strange. "Use ...
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1answer
29 views

Please help me clear confusion over principal roots and identities for n-th radicals

From my old high school math textbook: If ${a{\geq }0}$ and $n\in \mathbb{N} ^{\ast }$, then ${\sqrt[{n}] {a}}$ is the non-negative solution of ${{x}^{n}}=a$. It then goes on to infer a number of ...
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0answers
79 views

Integral representation of $f(x)=0^x$

Recently I had an argument with Luboš Motl on Quora, where he had argued that $0^0$ should be left undefined in computer algebra systems, because $x^y$ has no limit at $(0,0)$ and $0^x=0$ at all $x>...
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5answers
48 views

Converting into alternate e-Form

I was checking my results of an exercise online. The Maximum of my function was (1): $$\frac{4^{\frac{-1}{\log(2)}}}{\log^2(2)}$$ Now I wolframalpha tells me the alternate form is (2): $$\frac{1}{e^2\...
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3answers
161 views

Fractional exponentiation of functions

I run into a simple question about the fractional exponentiation of a function. Suppose that $f(x)$ is a real valued function which is always positive definite on its domain. Is it possible to obtain ...
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2answers
40 views

Finding what the sum of $(2^n)(3^{(-n+1)})$ from $n=1$ to infinity converges to [closed]

Sorry for the lack of math symbols but I'm trying to find what the sum of $f(x)=(2^n)(3^{-n+1})$ from $n=1$ to infinity converges to. I have tried using the sequence of partial sums but I'm too stupid ...
2
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1answer
160 views

Collatz Conjecture: Does it follow that since there is no $1$-cycle, if $ab > 1$, then $\frac{3^a - 2^a}{2^{a+b} - 3^a}$ cannot be an integer?

For me, one of the most interesting results of the Collatz Conjecture is that if an $m$-cycle exists, then $m > 68$. So, there are no non-trivial $1$-cycles, $2$-cycles, ..., up to $68$-cycles. ...
1
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1answer
114 views

If $ab > 1$, can $\frac{3^a - 2^a}{2^{a+b} - 3^a}$ be an integer?

I've been thinking about this positive fraction where $a,b$ are positive integers and whether it can be an integer for $ab > 1$: $$\frac{3^a - 2^a}{2^{a+b} - 3^a}$$ Clearly, if $a=1, b=1$, then: $$\...
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2answers
42 views

Euler's number inherently in $\mathbf{x^a=a^x}$ and $\mathbf{x.a=a^x}$?

I was solving some equations graphically, when I came across an identity of Euler's number in two cases: Case 1: $\mathbf{a.x = a^x}$ Ever wondered how for two numbers $\mathbf{\left\{{ a,x,y} \right \...
2
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1answer
51 views

sqrt of dB number

If the variance is $\sigma^2= 8 {\bf \, dB}$, and I want to calculate standard deviation $\sigma=?$ in dB. Which of the following is correct a. $\sigma = 8^{0.5} = 2.82 $ dB b. $\sigma =10\log_{10}( \...
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2answers
73 views

Golden Exponent? Tetration

I read somebody say “golden exponent $ x^x=x+1 $” now I didn’t understand what he meant but it really fascinated me thinking about some type of tetration version of the golden number. A number with a ...
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1answer
42 views

Complex large exponentiation

I have the next exponentiation of complex number $$z ={(1+ i\sqrt 3)}^{2020}$$ I was using the Theorem of Moivre but I got a $$2 ^ {2020}$$ Then how can I get this exponentation without computer ...
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0answers
50 views

Given $a,b,c$ solve $1=x^a + x^b - x^c$ for $x$

If $a,b,c$ are constants, is it possible to isolate $x$ in the following equation? $$1 = x^a + x^b - x^c$$ If so, how do I achieve this?

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