Questions tagged [exponentiation]

Question 1

In order to solve one exercise it would be helpful for me to have this inequality. I don't know if it's true or not so I checked it in Python for small $ k $ and it seems true and moreover it seems ...

Question 2

I’m currently looking into differences of powers. I found that 3^2-2^3 is 1, 3^3-5^2 is 2, 2^7-5^3 is 3, etc. My python code didn’t find a solution for 6. In fact, 6 was the only number it didn’t find ...

Question 3

For positive integers $a$ and $n$, when does $a^n+1$ have $n$ divisors? This was a natural question that popped up when I was investigating numbers of the form $a^n+1$, but surprisingly I can't seem ...

Question 4

Let $x$ be a real number. Consider the following expression: $$ \sqrt[3]{\frac{x^{3} - 3x + \left(x^{2} - 1\right)\sqrt{x^{2} - 4}}{2}} + \sqrt[3]{\frac{x^{3} - 3x - \left(x^{2} - 1\right)\sqrt{x^{2} -...

Question 5

I found this limit in a textbook for university admission. Compute $$\lim_{x \to \infty}{\left|\frac{\sin x}{x}\right|}^{\frac{\sin x}{x}}.$$ Everything seems simple, just substitute $$t=\frac{\sin x}{...

Question 6

Let $${\boldsymbol A} = \begin{pmatrix} 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & -1 \\ 0 & 1 & 0 & 1 \end{pmatrix}$$ then find $\boldsymbol A^n$ ...

Question 7

I cannot find a closed form solution for $x$ in $\dfrac{x^2 e^x}{e^x - 1} = k$ where $\{x,k \} \in \mathbb{R}^+$. I thought there might be a PolyLog solution, but apparently there isn't. This ...

Question 8

$8^x + 16^x = 2(25^x)$. The answer is clearly $0$, but myself and a couple others have not been able to find an elementary algebraic way to crack it, using properties of exponents, logs, etc. Grateful ...

Question 9

Let $G$ be a Lie group and $G_1$ its identity component (the connected component containing $1_G$). I want to understand the result that every $g \in G_1$ can be expressed as a product $g=\exp(X_1)\...

Question 10

I am comparing the variance sizes of two financial stochastic processes (related to fractional Brownian motion), simply using the method of taking differences. Ultimately, I need the roots of the ...

Question 11

I understand that $\log_1(1)$ is considered an indeterminate form, but the expression $\log_0(0)$ seems even more subtle. Algebraically, it is undefined because a logarithm cannot have a base of zero, ...

Question 12

Binary exponentiation provides an algorithm for calculating a power $a^n$, where you iterate over the binary digits of $a$, and at each step update $\mathbb{val} \leftarrow \mathbb{val} ^ 2$ $\mathbb{...

Question 13

I find myself studying equations of the type: $x^a+1=0$ for rational values of $a$ less than $1$, and so far I haven't been able to make much progress as to how to get to a proper solution, nor verify ...

Question 14

Let $r$ be a real number greater than or equal to $1$. I define the positive integer power sequence associated with $r$ to be the sequence of floors of $n^r$, where $n$ is a positive integer. For ...

Question 15

the problem Let the natural numbers $m>n \geq 1$. Show that for every $a \in \mathbb N^*$ there exists $b \in \mathbb N$ for which $a^m < b^n < (a+1)^m$. My idea: So we can also show that ...

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Questions tagged [exponentiation]

Proving $ \left(\frac{2^k}{2^k - 1}\right)^{k-1} \le 2 $ [closed]

Are there four whole numbers $a$, $b$, $m$, and $n$, where $a$, $b$, $m$, $n \ge 2$, and $a \ne b$, where $a^m-b^n=6$?

When does $a^n+1$ have $n$ divisors?

Is it possible to simplify a sum of these two cube roots by completing the cube?

Issue with a limit $\lim_{x \to \infty}{\left|\frac{\sin x}{x}\right|}^{\frac{\sin x}{x}}$

Find the $n$th power of a $4$ by $4$ matrix

Exponential and polynomial problem [closed]

Can $8^x + 16^x = 2(25^x)$ be solved algebraically? [closed]

Lie group element as finite product of exponentials

Is there a specific name or closed-form solution for equations of the form $a^x - b^x = c$? [duplicate]

Is $\log_0(0)$ undefined, indeterminate, or both? [duplicate]

Relationship between binary exponentiation and Horner's method evaluation of Robinson polynomials at $x=2$

Solving $ x^a+1=0 $ for rational $a$ less than 1

Does every $r \geq 1$ give a different power sequence? [closed]

Show that for every $a \in \mathbb N^*$ there exists $b \in \mathbb N$ for which $a^m < b^n < (a+1)^m$, where $m>n\geq 1$ are natural numbers.

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