# Questions tagged [exponentiation]

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

3,543 questions
Filter by
Sorted by
Tagged with
1k views

### Log power rule problem

According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian plane....
30 views

### How can I simplify $(n\,\mathrm{ mod }\,b^{k+1})\,\mathrm{ mod }\, b^k$? [duplicate]

I am trying to simplify an expression I've found that is related to converting from a number base to another: $$n\,\mathrm{ mod }\,b^{k+1} - (n\,\mathrm{ mod }\,b^{k+1})\,\mathrm{ mod }\, b^k$$ In ...
40 views

14 views

### Mean/median/etc of a zero-mean random variable with exponent

Suppose $\epsilon \sim N(0,\sigma^2)$, $a\in(0,1)$ and $x>0$ are constant. Is there any way of estimaing the following: $$(\frac{x+\epsilon}{x})^a$$ Although $\mathbb{E}[\frac{x+\epsilon}{x}]=1$, I ...
284 views

### Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$f : \mathbb R \to \mathbb R \text ,$$ $$f \left( a ^ b \right) = f ( a) ^ { f ( b ) } \text ,$$ when $a , b \in \mathbb R$, $a , b \ge 0$. So far the ...
41 views

### How to do exponential division when base and exponent is different between numerator and denominator?

How to do division when (a power x) / (b power y) Say, a,x,b,y are distinct values. For example, (2 power 19) / (7 power 23). Doing mathematical operations with bigger exponents is tedious and ...
47 views

### Definition of $x^{-a}$ and $x^{1/a}$

So, the other night I was wondering why $x^{-a}=\dfrac{1}{x^a}$ and $x^{1/a}= \sqrt[a]{x}$, because in school the teacher (at least for me), gave this formulas without any explanation whatsoever. This ...
5k views

### Why is $x^0 = 1$ except when $x = 0$?

Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
37 views

### I need to prove that $(a^n)(b^n) = (ab)^n,$ where $a,b\in\mathbb N$. Proof by induction on $n \in \mathbb Z, n\geq 0$.

I need to prove that $(a^n)(b^n) = (ab)^n,$ where $a,b\in\mathbb N$. Since $a^0=1$, is given in the question, I assumed my base step to be when $(n=0)$. $(a^0)(b^0)= (1)(1) =1.$ and now I'm stuck. I ...
65 views

27 views

### Why is $(e^{2πi})^i$ different from $(e^i)^{2πi}$ [duplicate]

I'm a high school senior and this question has been on my mind since 8th grade, since I've learned Euler's formula but I've thought about it a lot more recently. According to Wolfram Alpha: (e^i)^(2iπ)...
33 views

### Can you take a fractional root of a number?

Is it possible to say, take a 1,5th root of 2? And if not why so? Wouldn’t a 0,5th root of a to the third power = a to the power of 3/0,5 = a to the power of 6? If this is not allowed than why?
57 views

### For which positive integers $x$, $y$ satisfy the following equation: $x^2 + y^2 = 2020$?

This problem is driving me absolutely crazy. I managed to determine the max value of $x$ and $y$: $x^2 + y^2 = 2020$ $=>x^2 = 2020 - y^2$ It's obvious that square cannot be smaller than 0, and we'...
93 views

### Why don't we use subscript for roots?

I was wondering, why do we use a radical sign $\sqrt{}$ for roots? Why don't we use subscripts instead? It would make sense too, regarding the fact that $\sqrt{5^3} = 5^{3/2}$. A switch like that ...
3k views

### Prove that $a^x$ is continuous

I'm having trouble with proving the following: Let $a > 0$ be a positive real number. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) := a^x$ is continuous. I'm a ...
34 views

36 views

### Restrictions on exponential laws

Wanna be sure this is fully correct: So i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the laws i want to write ...
41 views

### Restrictions on exponential

This question has already been asked but no one answered so: so i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the ...
40 views

### Is it the case that there is some string, S, such that every string ending in S ends a perfect square? If so, is the same true for every power?

I am not a mathematician, but I gather from this paper that any string of digits ending in '1', '3', '7', or '9', no matter what it is, is the ending sequence of the decimal representation of ...
27 views

### Rewrite $\frac{2}{5}(x-2)^\frac{5}{2} + \frac{4}{3}(x-2)^\frac{3}{2} + c$ to $\frac{2}{15}(3x+4)(x-2)^\frac{3}{2} + c$

I must get the expression $\frac{2}{5}(x-2)^\frac{5}{2} + \frac{4}{3}(x-2)^\frac{3}{2} + c$ into $$\frac{2}{15}(3x+4)(x-2)^\frac{3}{2} + c$$ I tried expanding the $\frac{4}{3}(x-2)^\frac{3}{2}$ ...
7k views

### Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also,...
173 views

This question arose on the code golf StackExchange: Is there a solution to $2^a+2^b = 10^c+10^d$, with $0 \leq a < b$ and $0 \leq c < d$? In other terms: is there an integer that looks like $\... 2answers 76 views ### Exponents and Log properties This is part of a proof I am working on. So on one line, I needed to show that $$2^{\Bigg(\cfrac{\ln\frac{1}{n}}{\ln(2)}\Bigg)} =\frac{1}{n}$$ That was$2$to the power of$\ \ \cfrac{\ln\frac{1}{...
30 views

Alright, so i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the laws i want to write restrictions on. SO, please let ...