# Tagged Questions

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

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### Proof that $2^{2^x}$ ends in 6

So I just checked every number of the form $2^{2^x}$ up to $2^{8192}$ and they all end in $6$. Can someone formally prove that this will be true for all $x$?
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### formula for equation with exponent variable?

is there a closed formula for such an equation to find the value of $x$ in $ax = b^x$ if there isn't , are there any published attempts ?
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### Why is it more efficient to compute the modular exponentiation by calculating to the power of two and not three for example?

I learned about modular exponentiation from this website and at fast modular exponentiation they calculate the modulo of the number to the power of two and then they repeat this step. Why not ...
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### Prove that $\frac{p}{q}$ is a rational number with a finite decimal expression if $p$ is an integer and $q=(2^n)(5^m)$

Let $p,q$ be two integers and $q=(2^n)(5^m)$. Then $\frac pq$ is a rational number with a finite decimal expression. Any ideas how to do this? I've been thinking about it all day but I have no idea ...
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### First digits of a cube of a natural number

Can a cube of a number be of form: $2016a_1a_2a_3\dots a_n$? I have no direction, and would love to get a certain direction/proof. Thanks in advance
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### Axiomatic definition of exponent

Here's something I'm thinking about for a while, and would like to get feedback and some relevant references. Say I want to define by axioms an operation that will act like exponent, without using ...
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### Finding the solutions of $n! \ge n^a$

Let $a \in \mathbb{N}, a \ge 2$ be a fixed natural number. Consider the inequality: $$n! \ge n^a$$ It can be proven that this inequality is true for sufficiently large values of $n$, but how can we ...
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### Why is exponentiation right associative? [duplicate]

From Wikipedia: In order to reflect normal usage, addition, subtraction, multiplication, and division operators are usually left-associative while an exponentiation operator (if present) is ...
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### Expressing Factorials with Binomial Coefficients

Expression I have somehow stumbled upon this expression (I believe I have proved it, but that is not important right now), which I have tried to simplify by writing it like something like this (I ...
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### How is Faulhaber's formula derived?

I have been wanting to understand how to find the sum of this series. $$1^p + 2^p + 3^p +{\dots} + n^p$$ I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural ...
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### Proof of part of properties of exponentiation Tao proposition $4.3.12$

If you let $x,y$ be non-zero rational numbers, and let $n, m$ be integers, I need to prove that if $x \geq y>0$, then $x^{n} \geq y^{n}>0$ if n is positive, and $0< x^{n} \leq y^{n}$ if $n$ ...
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### Is a real logarithm of a special orthogonal matrix necessarily skew-symmetric?

The exponential map from the Lie algebra of skew-symmetric matrices $\mathfrak{so}(n)$ to the Lie group $\operatorname{SO}(n)$ is surjective and so I know that given any special orthogonal matrix ...
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### Unspecified $x^y$ vs. $y^x$ - which is larger?

Given only the expressions $x^y$ and $y^x$ and no additional information except $x\neq y$ (and the meta-knowledge that the problem was presented in the context of induction), is it possible to ...
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### Alternative to exponents

In a loose sense addition is repeated successorship, multiplication is repeated addition, and exponentiation is repeated multiplication. However, the latter is the only one that isn't well defined ...
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### Proving $a^b$ is well defined

How do I prove that $$\lim_{(m,n) \to \infty} a_m^{b_n} = a^b$$ where $a,b \in \mathbb R$, $a_i,b_i \in \mathbb Q$, $a_m \to a$, $b_n \to b$ and $a$ and $b$ are not both zero, and $a_m >0$ I can ...
### Does the complex modulus satisfy the power identity $|z^r|= |z|^r$?
Can we "split the modulus" of complex numbers? Let $z\in\mathbb{C}$. Then, does $$|z^{r}|=|z|^r$$ hold, where $r\in\mathbb{R}$. Is this true even for $r\in\mathbb C$ ? Also, can we show this? I am ...
The problem from the textbook is: Prove that if (a complex number) $z$ is a number on the unit circle, then $z$ has finitely many distinct powers $z^n$ if and only if the argument of $z$ is a ...