# 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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### 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 ...
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### 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 ...
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### 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 ...
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### Evaluating difficult integral with exponentials

I'm trying to evaluate a difficult integral. I'm able to break it down in separate terms and deal with scalar multiplication. However, I'm stuck trying to evaluate two terms in particular. Here is the ...
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### HowTo Rearrange The Piano Key Frequencies Formula

The Piano Frequencies formula maps the position of a piano key on the standard piano n to a specific note and frequency e.g. key ...
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### Exponentiation on a proof by induction

I am beginning to learn about proofs by induction, but cant get my head around the steps taken in a slide we've been shown. My problem lies within the red box, and I'm not sure if I'm missing some ...
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### 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)...
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### Partial proof by induction of the inequality : $(1-x)^{(2x)^n}+x^{(2(1-x))^n}\leq 1$

Claim Let $0.5\leq x<1$ and $n\geq 2$ a natural number then we have : $$(1-x)^{(2x)^n}+x^{(2(1-x))^n}\leq 1\quad (I)$$ Sketch\Partial (of) proof . We use a form of the Young's inequality or ...
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### 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?
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### 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'...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\frac{4}{3}(x-2)^\frac{3}{2}$ ...
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### Significance of $e^\pi$

I know that $$\sqrt{e^\pi}=\sqrt[i]{i}$$ which is quite a nice result, and that it is transcendental. Does the constant $e^\pi$ have any other significance? Thanks. I realize this is quite a trivial ...
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### The number of digits in an exponent of two

At the moment my area of research is Mersenne Primes. I am currently looking into a specifically large exponent, $\ 2^{112401533}-1$, and I can’t help but wonder how I would calculate how many digits ...
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### What is $-1$ to the power of a fraction?

If I have any negative number (does not have to be $-1$), how would I determine that value to the power of $1/2$? I learned that $x^\frac{1}{2}$ is equal to $√x,$ so wouldn't $-1^\frac{1}{2}$ be equal ...
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### Restrictions on exponent laws

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 ...
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### Understanding the evaluation of a fraction

Sorry if this is a little simple to ask here; but I'm at a loss as to where else to look for help with this one. I've been attempting to understand the evaluation of a fraction. The question I'm ...
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### How to solve for $x$ in the equation $y = (x + 25)^x$ where you know the value of $y$

I am trying to figure out how to solve for $x$ in the equation $y = (x + 25)^x$ for any known $y$. For example, I know that in the example $2758547353515625 = (x + 25)^x$, the value of $x$ is 10. Is ...
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### With what exponent rules does $-(2^{-55} + .4\times2^{-56})+(2^{-54})$ become $.1\times2^{-52}$?

I have a math class focusing on numerical analysis so I'm working with very small numbers. My professor has set $0.4 − 2^{−55} − 0.4×2^{−56} + 2^{−54} = 0.4 + 0.1×2^{-52}$ but shown no steps in ...
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Okay, being completely honest, i don't know how more to make it clearer, this question has been deleted 3 times, maybe people don't actually read what i say at the beginning, which said perfectly what ...
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### Is $(\ln{x})^a$ written as $\ln^a{x}$? [duplicate]

This is a basic notation question. With trig functions like $\sin{x}$ and $\cos{x}$, if they are raised to a power, let's say $a$, then we write it, in the case of $\sin{x}$, as $\sin^a{x}$. However, ...
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### What can be a generalization of repeats in exponentiation using modulo?

I came across a Math Problem in a Japanese Coding Test(It is officially over now so no worries about discussing it, https://atcoder.jp/contests/abc179/tasks/abc179_e). I will write the mathematical ...
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### Restrictions on laws

I'm wondering about the restrictions, my doubt is for example at $\log_a(b)=c\implies a^c=b$, how would anyone add the restrictions for this? I know the argument and the base of a log have to be >0 ...
### Finding roots : $x-7\sqrt{x}+10$
Find all possible roots of $k(x) = x-7\sqrt{x}+10$ I am having serious trouble rearranging this function as $ax^2+bx+c$ since it has $'-7\sqrt{x}'$. Can anyone please help me? Little help would ...