# 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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### EGMO 2014/P3 : Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist ...
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### why $\left(\left( \left(-\frac{1}{4}\right)^{-2}\right)^\frac{1}{4}\right) \neq \left(\left(-\frac{1}{4}\right)^{-\frac{1}{2}}\right)$?

Why $\left(\left( \left(-\frac{1}{4}\right)^{-2}\right)^\frac{1}{4}\right) \neq \left(\left(-\frac{1}{4}\right)^{-\frac{1}{2}}\right)$? well, I tried this question but as far my calculations, I am ...
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### Continuity of $f(t)=t^{x-1}$

In our material we have the statement that for a given $x\in\mathbb{R}$ with $x\geq 1$ the function $f(t):[0,1]\to \mathbb{R}$, where $f(t)=t^{x-1}$, is continuous (if we define $0^0:=1$). Then a few ...
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### Interpretations of Exponents

I have been reading one of the proofs of Euler's identity, $e^{i\theta}=\cos(\theta)+i\sin(\theta)$. I have always thought that exponents can be interpreted as its base being multiplied its exponent ...
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### Solution to : $g(x)=(\sin^2(x))^{\frac{1}{f(x)}}+(\cos^2(x))^{f(x)}\leq 1$

Well, this is related to my own question Pretty conjecture $x^{\left(\frac{y}{x}\right)^n}+y^{\left(\frac{x}{y}\right)^n}\leq 1$ . Let $x\in(-\infty,\infty)$ and define $g(x),f(x)$ continuous and ...
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### How do I find the value of $\operatorname{LCM} (a_1^k,a_2^k, \ldots ,a_n^k) \pmod {m}$?

LCM can be upto or greater than $10^{50}$. $k$ and $m$ are upto $10^9$. My current approach is Finding the LCM . (LCM % m)^k=A. A % m. This is working but takes a long time in finding LCM using GCD. ...
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### How to prove that $(a^m)^n=a^{mn}$ where $a,m,n$ are real numbers and a>0?

I know how to prove the equality when $m$ is a rational number and $n$ is an integer, but do not know how to go about proving this for real numbers. On a semi-related note, I was trying to prove this ...
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### Is there a name for this softmax-like equation?

I am working on some software and have a need to normalize a list of values. I originally was using a typical softmax, but found that it had some undesirable properties for what I was doing. I wound ...
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### Exponential growth and decay,and the constant k

In the equation $$A=Be^{kt}$$,where $B$ is the initial amount and $t$ is the time taken what is $k$,I know it's a constant of proportionality ,but is it the same as the number of time a certain amount ...
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### Prove that there are infinitely many primes of the form $2^n-n^2$

My original question is: Find all the number $x$ such that, for all $n \in \mathbb{N}, n\ge 5$, we have $2^n-n^2 | x^n -n^x$. The first thing to notice is that, if $n$ is even, then $2^n-n^2$ is even ...
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### I don't understand this exponent simplification

I've been doing the Khan Academy math courses to brush up on my math foundations before starting my CS/math degree in the fall semester. I just don't fully understand negative exponents, I stumbled ...
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### Is there a term for the function $\operatorname{sign}(x) \times |x|^a$?

I wanted to use a power function for some procedural generation, but also needed to preserve the sign of the input and not be limited to uneven integer exponents. This seems like a basic enough use ...
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### Exponential handling

I saw this equation somewhere: $e^(- Ef/kt)$ = $e^(- Ev1/kt)$ + 0.5$e^(- Ev2/kt)$. The author then ends to this: $e^(- Ef)$ = $e^(- Ev1)$ + 0.5$e^(- Ev2)$. All the kt are removed. I did not ...
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### Solve : $\Bigg(\frac{ab}{a^{2a}+b^{2b}}\Bigg)^{a^n2^{n-1}}+\Bigg(\frac{ab}{a^{2a}+b^{2b}}\Bigg)^{b^n2^{n-1}}\leq 1$

Let $a,b>0$ such that $a+b=1$ and $n\geq 2$ a natural number then we have : $$\Bigg(\frac{ab}{a^{2a}+b^{2b}}\Bigg)^{a^n2^{n-1}}+\Bigg(\frac{ab}{a^{2a}+b^{2b}}\Bigg)^{b^n2^{n-1}}\leq 1$$ We have ...
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### Irrational Number modulo Prime

Given three positive integers $a$,$r$ and $n$, I need to calculate $\left\lfloor(a+\sqrt{r})^n\right\rfloor$ mod $p$, where $p$ is a prime. Let $\left\lfloor(a+\sqrt{r})^n\right\rfloor=c+d\sqrt{r}$. ...
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### Proving exponent law for real numbers using the supremum definition only

I was working on a problem displaying the expansion of the definition of exponents, and naturally the final question was to prove the exponent laws when the exponents are real numbers. For $a>1$, ...
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### How does this differential equation's solutions embedded in an exponential-exponential coordinate plane change the form of the differential equation?

$u=\log(x)$ and $v=\log(y)\implies uv=k.$ $uv=k$ satisfies an algebraic differential equation. $X=(x,-y)$ is a vector field with integral curves $uv=k$ in $u-v$ coordinates. A differential equation is ...
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### If $x \in \mathbb{Q}$ and $b \in \mathbb{R}$ with $b > 1$ and $x > 0$, then $b^x \geq 1$ [closed]
The following problem is motivated from Baby Rudin, Chapter 1, Problem 6(c): if $x \in \mathbb{Q}$ with $b > 1$ and $x \geq 0$, then $b^x \geq 1$. I would prefer not to use techniques such as ...