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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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How to construct exponentiation over all of $\mathbb{R}$?

I want to construct exponentiation over $\mathbb{R}$. $$^\wedge:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$$ Using a strictly constructive approach $\mathbb{R}=\{[(x_n)]|(x_n)\in\mathbb{Q}^\mathbb{N}\}$, ...
Isaac Sechslingloff's user avatar
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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 ${...
Nerses Asaturyan's user avatar
-3 votes
0 answers
62 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,...
PortyMart's user avatar
5 votes
1 answer
162 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 ...
user158293's user avatar
-3 votes
2 answers
126 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 ...
pdh0710's user avatar
  • 103
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0 answers
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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 ...
ca_100's user avatar
  • 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. ...
Đỗ Quốc Khánh's user avatar
-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 ...
IONELA BUCIU's user avatar
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 ...
Poverlord's user avatar
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 $3^{x} \bmod 21$? $3^0 = 1$ $3^1 = 3$ $3^2 = 9$ $3^3 = 6$ $3^4 = 18$...
zakrea2070's user avatar
-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 ...
Jkt's user avatar
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1 vote
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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{...
Hussain-Alqatari's user avatar
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 ...
Shyam's user avatar
  • 49
1 vote
2 answers
51 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 $M$ ...
Frederik Ravn Klausen's user avatar
1 vote
1 answer
48 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 ...
jared soto's user avatar
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0 answers
34 views

Why is there no logarithmic form of the exponential distributive rule/power of a product rule?

When learning the laws of exponents and logarithms, one finds that there is a correspondence. Each law of exponents has a corresponding equivalent expression in terms of logarithms. For example, the ...
ziggurism's user avatar
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2 votes
1 answer
66 views

Closed form for $\sum \left (\pm a_1 \pm a_2 \pm \dots \pm a_n\right )^\ell$

I realized that if you take the $2^n$ quantities $$\pm a_1 \pm a_2 \pm \dots \pm a_n$$ and consider the sum of their squares, then the product terms cancel out nicely to give $$\sum \left (\pm a_1 \pm ...
Dumbest person on earth's user avatar
1 vote
3 answers
102 views

If $f(x)=m(x) + \mathcal O\left( e(x)\right)$, then is $f(x)^\ell = m(x)^\ell + \mathcal O\left( e(x)^\ell\right)$?

I am new to advanced calculus and I am a bit confused about the following. If I know that a function $f$ varies as $$f(x)=m(x) + \mathcal O\left( e(x)\right)$$ then is it okay to say that $$\left(f(x)\...
Dumbest person on earth's user avatar
5 votes
3 answers
164 views

Solve for $x$: $4^x + x = 11$

I came across this question on Facebook (as a puzzler): $\begin{align}\\ 2^x + x = 11\\ \end{align}$ I solved this by using the following: Since $2^x$ is even, therefore $x$ must be odd. So $\exists ...
ewokx's user avatar
  • 496
1 vote
0 answers
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Simplifying two rational expressions with roots

Can you please check the steps I followed to simplify the following two expressions? Given that $0 < n < 1$, simplify: $$\left( \frac{\sqrt{1+n}}{\sqrt{1+n} - \sqrt{1-n}} + \frac{1-n}{\sqrt{1-n^...
Aleksandar Živković's user avatar
1 vote
4 answers
923 views

Why roots aren't the inverse of exponentiation but logarithms?

I think it's easy to see it when we look at the inverse of the function "$f(x) = a^x$" but I wonder if there's other way to look at it besides just analyzing the function. I was taught my ...
pingu's user avatar
  • 21
1 vote
0 answers
41 views

Is there a Relation for Exponentiation Similar to $\leq$ or $\backslash$ (divisibility)?

I'm trying to define common mathematical options on the natural numbers. I am doing this because operations like subtraction are commonly constructed as just the addition of the inverse, however this ...
Isaac Sechslingloff's user avatar
1 vote
0 answers
47 views

Expectation of real number exponentiated by Poisson process

Consider a Poisson process $N_t$ with intensity $\lambda>0$ and let $x$ be a real-valued number. In principle, from the properties of the Poisson distribution, we have: $$\mathbb{E}(x^{N_t})=e^{\...
Morris Fletcher's user avatar
2 votes
1 answer
62 views

second power in exponential

I am struggling with raising some powered expression to another power, like $(a^b)^c$ which should equal to $a^{bc}$. As two examples $(3^2)^3=9^3=729=3^6$ $3=3^{2/2}=(3^2)^{1/2}=(3^{1/2})^2$. Now if ...
Shoaib Mirzaei's user avatar
5 votes
3 answers
90 views

How to prove if $e<a<b$ then $a^b>b^a$

How to prove if $e<a<b$ then $a^b>b^a$ Thus far I got: $a^b>b^a$ $e^{\ln(a^b)}>e^{\ln(b^a)}$ $\frac{e^{b\cdot \ln(a)}}{e^{a\cdot \ln(b)}}>1$ $e^{b\cdot \ln(a)-a\cdot \ln(b)}>1$ $b\...
Jurre van Iersel's user avatar
0 votes
0 answers
47 views

Why is the formula for compound interest the way it is?

I understand the compound interest formula when it only involves compounding yearly at a certain rate, but I do not understand the concept of compounding within a year. For example, if the investment ...
Rohan Rajasekar's user avatar
2 votes
3 answers
109 views

Can we call $\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{d}\right)}=\frac{a d}{b c}$ a Law of exponents?

Can we call $\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{d}\right)}=\frac{a d}{b c}$ a Law of exponents? Sorry if my question is silly. I proved this using the well known laws of exponents $\frac{...
LifeIsMath's user avatar
2 votes
0 answers
75 views

Principal branch of $z^{1-i}$

I am solving a problem Find the principal branch of $z^{1-i}$. I wanted to verify my solution. I know we can write $z^{1-i} = e^{(1-i)\cdot \text{Log}_e(z)}$ Since the principal branch of $\text{Log}...
A. Srivastava's user avatar
-2 votes
2 answers
115 views

what is the definition of $e^z$ in 4.7 Trigonometric functions, Terence Tao's Analysis 2

In Terence Tao's "Analysis 2", Section 4.6 "A digression on complex numbers" meticulously defines $$\exp(z):=\sum_{n=0}^\infty\frac{z^n}{n!}$$ for a complex $z$. However, in the ...
ZhanYi's user avatar
  • 3
3 votes
2 answers
77 views

How to evaluate an expression of higher powers and roots using logarithms?

I am struggling with the following question from a Dutch algebra exam from the 1950s. The instructions are as follows: Calculate with logarithms. $$ x = \frac{\sqrt[3]{(23.57^2 - 15.63^2)}}{{0....
Marinus Lilienthal's user avatar
5 votes
5 answers
358 views

Which 'laws of indices' hold for complex bases and real powers?

Specifically, I would like to know whether: $$z^a z^b = z^{a+b}$$ $$(zw)^a = z^a w^a$$ $$(z^a)^b = z^{ab}$$ hold true for any and all $z,w \in \Bbb{C}$ and $a,b \in \Bbb{R}$? Also, I'd be interested ...
Anis Manuchehri-Ramirez's user avatar
-2 votes
1 answer
59 views

How does $\log(y)=C+t$ become $y = C e^{t}$? [closed]

I came across this transformation : $$\begin{align} \log(y) &= C + t \tag{1} \\[4pt] y &= C e^{t} \tag{2} \end{align}$$ How was the first step simplified into the second?
codeman's user avatar
-1 votes
3 answers
110 views

$2^{\cos^2(x)}=\sin(x)$ [closed]

Is there an algebraic way to solve $$2^{\cos^2(x)}=\sin(x)?$$ I tried $$\cos^2(x)=\log_{2}(\sin(x))$$ but I have no idea how it could help. Also, since $\cos^{2}(x)=1-\sin^2(x)$, then $$\frac{2}{2^{\...
mvfs314's user avatar
  • 2,084
4 votes
1 answer
178 views

Do iterated factorials or iterated exponents grow faster?

This is purely out of curiosity and I'm not quite at the point in calculus where I know how to prove either for myself... Given $$n^{n^{n^{n^n}}}$$ and $$(((n!)!)!)!$$ As n approaches infinity, which ...
æ æ's user avatar
  • 86
8 votes
2 answers
755 views

Algebra Math Olympiad Question

The following question is a Math Olympiad Problem: Find $x>0$ that solves $x\sqrt{x\sqrt x}=2$ The answer is $\sqrt[7]{16}$ but I got $2\sqrt[3]{2}$ by: $$\begin{align}x\sqrt{x\sqrt x}=2&\to \...
Kcharliee's user avatar
0 votes
2 answers
70 views

Suppose a colony of cells starts with 10 cells, and their number triples every hour. After how many hours will there be 500 cells?

I thought it would be log(500), which gives approximately 2.69897. I know that there could be alternative forms of the answer, but for the life of me, I don't understand how they arrive at this ...
David A.'s user avatar
0 votes
0 answers
85 views

Paradox $1<1$ using complex exponentiation [duplicate]

\begin{align*} 1&>e^{-4\pi^2} \\ &= e^{(2\pi i)(2\pi i)} \\ &= (e^{2\pi i})^{2\pi i} \tag{$\ast$} \\ &= 1^{2\pi i}\\ &= 1 \end{align*} The problem with this fake proof is that ...
J P's user avatar
  • 893
-2 votes
1 answer
52 views

Why does power of power yield different results [closed]

$2^{(2x+1)^{0.5}}=32$ or $2^{\sqrt{(2x+1)}}=32$? Scenario 1 $2^{(2x+1)^{0.5}}=2^5$. $2^{0.5(2x+1)}=2^5$ (power of power rule) $\log2^{0.5(2x+1)}=\log2^5$ (log both sides, pull the exponents out front,...
Recramorcen's user avatar
0 votes
1 answer
20 views

Analytical solution to a system of equations with exponentials relating to dry heat sterilization

For my job, I sometimes deal with lethality values for units equipped with dry heat sterilization cycles. We run one of these cycles and obtain temperature readings at a given location within the unit ...
viscous_cat's user avatar
0 votes
1 answer
65 views

What is the term for when the power is written behind the base?

Some time ago I watched a video discussing the notation of a “backwards” exponent, where the power comes before the base (e.g. ³2). I was wondering: a. What this term was called b. How it works Thanks
Selisine's user avatar
1 vote
6 answers
161 views

How to approximate $1.05^{50}$ by hand

Is there some type of Taylor expansion or something which I could use to approximate quickly what for example $1.05^{50}$ is? or put bounds on that number? It's really annoying because I can't even ...
Bobas_Pett's user avatar
0 votes
1 answer
125 views

Quickly putting something to the power of 100 without a calculator (Shortcuts)

For some probability questions, I was wondering if anyone knew of any tricks on how to do do the following; let's say we have a fraction of 49/50 and we want to put this to the power of 100 quickly ...
Julien Maas's user avatar
4 votes
7 answers
181 views

Compare $3^4 \times 6^5 \times 7^8 \bigcirc 4^3 \times 5^6 \times 8^7$

Compare $$3^4 \times 6^5 \times 7^8 \bigcirc 4^3 \times 5^6 \times 8^7$$ Options: $\text{(A)} > \space\space\space\space\space \text{(B)} <\space\space\space\space\space \text{(C)} =$ Notes: $1....
Hussain-Alqatari's user avatar
0 votes
1 answer
51 views

What are the x intercepts (wherever defined) for $ \vert x^2 -8x +15 \vert ^ {(x-1)(x-2)(x-3)(x-5) \over (x-2)} = 1 $

The original question asked was the following : (Source : https://drive.google.com/file/d/1U7khRKh12GlaSY73210oQd2170JiXEGY/view) Rita took some of her friends for picnic. Her friends are at x−...
Devanshu Kashyap's user avatar
10 votes
6 answers
232 views

How to estimate $10^{\frac{1}{2}} + 10^{\frac{1}{3}} + \ldots + 10^{\frac{1}{10}}$

I heard an interesting interview question recently, which was as follows: Estimate the value of $X = 10^{\frac{1}{2}} + 10^{\frac{1}{3}} + \ldots + 10^{\frac{1}{10}}$. You have 30 seconds to Compute ...
Christopher Miller's user avatar
0 votes
1 answer
85 views

Generalization of the matrix exponential [closed]

I've seen this post which addresses the question of exponentiating a vector. I was wondering if there's a well-defined notion of exponentiating a rank $r$ tensor? For instance, if I have a rank 3 ...
confusion's user avatar
0 votes
3 answers
93 views

Why does $x^{\frac{3}{4}}$ have a domain of $x≥0$, while $x^{\frac{6}{8}}$ has a domain of $x\in{\Bbb{R}}$? [duplicate]

Why does the simplified version of $x^{\frac{6}{8}}$ have a domain of $x≥0$ while the unsimplified has a domain of $x\in{\Bbb{R}}$? Shouldn't they have the same domain being that they are the same ...
Max Heppelmann's user avatar
0 votes
0 answers
43 views

Question about exponent laws [duplicate]

I can't seem to demonstrate how $\frac{\sqrt{\sqrt{3}+2}}{2}$ is equal to $\frac{\sqrt{2}+\sqrt{6}}{4}$... even though my calculator tells me that both are equal $0.965925826$. Hoping someone can show ...
sesandc3123's user avatar
0 votes
0 answers
12 views

Calculate change of one variable based on relationship with other variable

If the volume scales V (km^3) with the area A (km^2) by a relationship V = 0.67*A^(1.262). Then how would the volume change if an area of 1161 km^2 decreases by 210.76 km^2?
Yoni Verhaegen's user avatar
0 votes
2 answers
44 views

Sum of numbers in $[0,1]$ raised to exponents at least as large as $1$.

For $a,b\in [0,1]$ and $\epsilon\geq 0$, does the following equality hold? $a^{1+\epsilon}+b^{1+\epsilon}\geq |a-b|^{1+\epsilon}$ All I can think to do so far is: \begin{align*} |a-b|^{1+\epsilon} ...
atul ganju's user avatar

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