Questions tagged [exponentiation]
Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.
4,359
questions
-3
votes
1
answer
76
views
Where does $x\mapsto e^{-x}-(1-\frac xn)^n$ reach its maximum? [closed]
For all integer $n\ge2$, the map $f_n:[0,n]\to\mathbb{R},x\mapsto e^{-x}-(1-\frac xn)^n$ reaches its maximum at some $x_n$. I could prove that $1<x_n<2$.
I would like to show that the sequence $(...
- 7,642
0
votes
0
answers
27
views
Is the product of exponentiated elliptic curve basis elements invariant under FFT of scalars?
I am working with an elliptic curve defined over a finite field $\mathbb{F}_p$ and have a basis set of points ${g_0, g_1, \ldots, g_n}$. When I perform the FFT on these points, I obtain a new basis ${...
-3
votes
0
answers
61
views
Trying to prove $(a^x)^y=a^{xy}$ using definition
In a lecture, we learned that $a^x$ is defined to be the limit of $a^{x_n}$ where $x_n$ is a series of rationals that converge to x. I was trying to prove $(a^x)^y=a^{xy}$ using solely this definition,...
- 41
5
votes
1
answer
159
views
Prove/disprove that $a^{2m} + b^{2m} + c^{2m} > 2^{1-m}$ subject to $a + b + c= 0$ and $a^2 + b^2 + c^2 = 1$
Problem. Let $a, b, c$ be reals with $abc\ne 0$, $a + b + c = 0$, and $a^2 + b^2 + c^2 = 1$. Prove or disprove that
$a^{2m} + b^{2m} + c^{2m} > 2^{1-m}, \forall m\in \mathbb{Z}_{>2}$.
Prior ...
- 415
-3
votes
2
answers
123
views
The value of $\sqrt{(-1)^2}$? [duplicate]
Which is the correct value of $\sqrt{(-1)^2}$ ?
$\sqrt{(-1)^2} = \sqrt{(-1)\cdot(-1)} = \sqrt{1} = 1$
$\sqrt{(-1)^2} = ((-1)^2)^{1/2} = (-1)^{2\cdot1/2} = (-1)^1 = -1$
I was surprised to realize ...
- 103
0
votes
0
answers
44
views
Can power enter modulus
can we say that
$|x|^n = |x^n|$ $ \forall n,x \epsilon $$R$?
for Real Numbers,
There should not be a problem since in place of x there can either be a +ve or a -ve number (since above ...
- 199
-4
votes
0
answers
44
views
Find $n^ 1 +n^2+\dots. + n^k$? [duplicate]
How do I find $n^ 1 +n^2+\dots. + n^k$?
I found a post that asks about $n ^ 1 + n ^ 2 +\dots+ n^{n - 1}$, but I only want until $n^k$, and I can't apply the answer of that post to fit in my use.
...
-1
votes
1
answer
45
views
solution-verification | Find $x$ from the some equalities
the problem
Find $x$ from the equalities:
a) $(3-2\sqrt{2})^x=3+2\sqrt{2}$
b) $(\sqrt{3}-2)^x=7-4\sqrt{3}$
c) $(5+2\sqrt{6})^{x^2-2x}=5-2\sqrt{6}$
my solution
a) $(3-2\sqrt{2})^x=3+2\sqrt{2} $ and ...
- 724
1
vote
4
answers
73
views
Highschool Math Problem About Exponentials $\frac{10}{1-10^{x-y}}+\frac{10}{1-10^{y-x}}$
So the problem goes:
Solve
$\frac{10}{1-10^{x-y}}+\frac{10}{1-10^{y-x}}$
I tried rationalizing but it got really complicated and I couldn't do much with it...
The answer is 10.
I hope someone can ...
- 13
0
votes
2
answers
72
views
What is the order of $p_{1}^{x} \bmod{n}$ where $p_1$ is a prime factor of $n$ [closed]
I am looking for a formula, algorithm, or even literature on the topic.
Take $21$ for example
$21 = 7 \cdot 3$
What is the order of $3^{x} \bmod 21$?
$3^0 = 1$
$3^1 = 3$
$3^2 = 9$
$3^3 = 6$
$3^4 = 18$...
- 19
-2
votes
0
answers
141
views
Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]
So the question was
$$\sqrt{x+1}=-2$$
And obviously there is no value for it,
However,
If you do the thing with $e$ and $\ln{}$
$$e^{\ln{\sqrt{x+1}}}$$
and
$$e^{\frac{1}{2}\cdot (\ln{x+1})}$$
Then ...
- 19
1
vote
0
answers
58
views
Finding $\displaystyle\sum_{k=1}^{5}k^{99}\pmod{5}$ [duplicate]
This problem is from a great book:
$$\color{rgb(128,128,255)}{\text{Discovering Higher Mathematics}}\\\color{rgb(255,128,128)}{\text{Four habits of Highly Effective Mathematicians}}\\\text{By }\color{...
- 5,450
2
votes
2
answers
62
views
Why the property of exponents holds true even for fractional powers
How can we prove that the property of exponents a^m × a^n =a^(m+n) (where "^" this sign denotes the power a is raised to)holds true even if m, n and a are fractions? Like I can clearly see ...
- 49
1
vote
2
answers
50
views
Examples of expansions of the exponential of a sum of two matrices [closed]
The exponential function of a matrix is fundamental in mathematics, physics and beyond.
One can define it using the power series
$$
e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}
$$
For any matrix $M$ ...
1
vote
1
answer
47
views
Properties of Nth roots and fractional powers
Context: I'm programming an arbitrary precision math library and created some weird algorithms to calculate a number raised to non-integer powers due to optimizations.
From my understanding, raising ...
- 13