# Tagged Questions

Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

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### Let $H = \{2^m : m \in \mathbb{Z}\}$ & define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rationals by $a\mathbin{R}b$ iff $a/b \in H$.

Let $H = \{2^m : m \in \mathbb{Z}\}$ and define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rational numbers by $a\mathbin{R}b$ if and only if $a/b \in H$. Prove that $R$ is an equivalence ...
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### Prove that $24^{31}$ is congruent to $23^{32}$ mod 19.

According to my knowledge, to prove that $24^{31}$ is congruent to $23^{32}$ mod 19, we must show that both numbers are divisible by 19 i.e. their remainders must be equal with mod 19. Please correct ...
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### Help solving the inequality $2^n \leq (n+1)!$, n is integer

I need help solving the following inequality I encountered in the middle of a proof in my calculus I textbook: $2^n \leq (n+1)!$ Where $\mathbf{n}$ in an integer. I've tried applying lg to both ...
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### Unprovable identity over the integers

I was thinking about Tarski's problem, and was wondering what happens if we have a theory $T$ with two sorts $N,Z$ with intended interpretations $\def\nn{\mathbb{N}}$$\def\zz{\mathbb{Z}}$$\nn,\zz$ ...
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### Powers with complex/negative bases

If x can be a positive real number (for example a fraction with a numerator and denominator), then why does the following relationship hold true only if and only if a and b are strictly positive real ...
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### Limit with logarithm: $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n}$

What is the limit $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n }$ (ln=natural logarithm) for alfa real and less than zero? I found out it is zero for $\beta\ge0$, since then you can use the ...
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### Find $x$ in the equation $x^x = n$ for a given $n$

Simply: How do I solve this equation for a given $n \in \mathbb Z$? $x^x = n$ I mean, of course $2^2=4$ and $3^3=27$ and so on. But I don't understand how to calculate the reverse of this, to get ...
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### How many positive integers x and y satisfy the equation $x^y = (2016)^{2016}$?

How many positive integers x and y satisfy the equation $x^y = (2016)^{2016}$ ? Explain your answer. I started by factoring $2016$. I found the factors to be $36$, but I couldn't go further.
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### Fractional Exponents - Is the sign discarded?

For example, 16^(3/4) Is the accepted as both -8 and 8 or just 8? I ask this because on an AS maths mark scheme it says to condone -8 Thanks
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### What does $x^{(i)}$ Mean or Denote

I know this is a simple question, but what does $x^{(i)}$ mean (where $x$ and $i$ are variables and $i$ isn't $\sqrt{-1}$) or what operation does it denote? I assume it's not a regular exponent. I saw ...
Let $N$ be the product of two primes and let $P$ be the smallest prime larger than $N$. Let the algorithm $R(N,s)$ return $s^{1/P} \pmod{N}$. Let the algorithm $\widehat{R}(N,s)$ pick a ...