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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2
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4answers
71 views

Can the 0 power rule be conceptualized?

A review of the rule that $n^0$ is always 1, when n is not 0, has made me question if all math rules can be visualized or conceptualized with a general intuition gained from viewing the natural world. ...
0
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3answers
48 views

What is the smallest value of n such that an algorithm running at 100*n^2 operates faster than 2^n ? [How to figure out without brute force]

Okay, so I needed to find the smallest value of n such that algorithm 100*n^2 is faster than 2^n. [what I have tried] So, I instantly thought '0'. But, I then realized it can't be 0, 0 implies that ...
1
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1answer
44 views

Numerically stable evaluation of $x^{n!}$

Given that $x$ is a real number with property $0 < x < 1$ and $n$ upto $4000$ Is there a good way to decompose the n! into steps for multiplying x?
1
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0answers
57 views

Intuitive understanding of Euler's Formula

Thanks for reading! In this link... https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ ...Euler's formula is explained pretty well. I understand that multiplying a ...
1
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2answers
43 views

A 1968 AHSME problem with exponents

Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are: $\text{(A) } x,z,y\quad \text{(B) } x,y,z\quad \text{(C) } y,x,z\quad \text{...
2
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2answers
68 views

Does this rule I found really work?

I was playing a bit whit exponents. I maybe found a working formula for calculating $n^y$ if you know $n^x$. The formula may already be discovered, but the formula I found is: $$ (n^x)^\frac{y}{x} = n^...
1
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5answers
69 views

How to Solve for $x$ in a Particular Exponential Equation

I am trying to solve for $x$ in $x^2=(16)^{2x}.$ So I started this way: I took square root of both sides and got $x=16^x$ Then I took the logarithm of both sides and got $\log x=x \log 16.$ This ...
0
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1answer
44 views

Dividing exponent with same base

There are $~2^{80}~$ possibilities to calculate and I want to divide it by $~2~$ to process it by two computers at the same time to find the answer maybe sooner. How can I divide $~2^{80}~$ by $~2~...
0
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1answer
44 views

Approximate $e^{1763.02}$ (in decimal notation) using only a standard scientific calculator

My attempt: $\log(e^{1763.02})=1763.02\log e\approx 765.66$ So, $e^{1763.02}\approx 10^{765.66}$ $\implies 10^{765}<e^{1763.02}<10^{766}$ $\implies 1\cdot 10^{765}<e^{1763.02}<10\cdot ...
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2answers
13 views

Equation for slope of exponential function of arbitrary base

Say I have some general exponential function defined as such; $$f(x,a,b) = a\left(1-b^{-x}\right)$$ Given any arbitrary value for $a$, how can I solve for $b$ such that $\frac{d}{dx}f(0,a,b) = 1$? ...
1
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1answer
34 views

More Clarification on the number of $I$th powers modulo $p$

Jack provides an answer and explanation as to how many $I$th powers there are modulo a prime $p$. However, I am unsure as to the logic behind the following step: When we apply the map $x\mapsto x^...
5
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5answers
99 views

Intuitively, why are the two limit definitions of $e^x$ equivalent?

Thanks for reading! Intuitively, why does... $$\lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{xn}=\lim_{n \rightarrow \infty} \left(1+\frac{x}{n}\right)^{n}=e^x$$ Note, I'm not asking why $...
1
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4answers
38 views

I have a simple problem which gives two different solutions in two different calculators.

I hope that this kind of questions doesn't break the site's rules. I have this simple problem which gives two different solutions in two different calculators (Wolfram Alpha and Symbolab). What am I ...
3
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5answers
134 views

Method to find colossally abundant numbers?

Colossally abundant numbers are positive integers $n$ for which there exists a positive exponent $\epsilon$ such that $$\frac{\sigma(n)}{n^{1+\epsilon}}>\frac{\sigma(m)}{m^{1+\epsilon}}$$ for all ...
5
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3answers
83 views

$n^n<(n-1)^{n+1}$ for integer $n\ge5$

For natural number $n\ge5$, by mathematical induction or otherwise, prove that $n^n<(n-1)^{n+1}$. Actually I was trying to solve the problem that I posted. My Attempt: Step I: Verify that when ...
8
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6answers
253 views

compare $m=50^{50}$ with $n=49^{51}$

A multiple choice question: If $m=50^{50}$ and $n=49^{51}$, then (A) $m>n$ (B) $m<n$ (C) $m=n$ (D) The given information is not enough My attempt: Since ordinary ...
0
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2answers
25 views

Given $ax^n = 2^k$ is it possible to calculate $k$ if $a$, $x$ and $n$ are known

For example, we know that: $2.143483648 * 10^9 = 2^{31}$ If we were given: $2.143483648 * 10^9 = 2^k$ How do we find $k$ ?
3
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3answers
47 views

How do you prove $\prod_{i=0}^n(1+2^{2^i})=2^{2^{n+1}}-1$ using induction?

For a homework assignment, I am being asked to find and prove (by induction) a simple formula for $\prod_{i=0}^n(1+2^{2^i})$. Plugging in a few small values of $n$ shows that the formula is $2^{2^{n+1}...
4
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4answers
91 views

If $a^b$ has $b$ digits, what is the greatest value of $b$?

For natural numbers $a$ and $b$, what is the greatest value of $b$ so that $a^b$ has $b$ digits? I knew that the greatest value of $b$ is $21$, where $9^{21}=\underset{21 \text{ digits}}{\underbrace{...
-1
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2answers
47 views

Why Does $a^0 = 1$ and $a^{-p} = {\frac{1}{a^p}}$ if $(a \not = 0) , p \not= 0$? [closed]

Why does $a^0 = 1$ and $a^{-p} = {\frac{1}{a^p}}$ if $(a \not = 0)$ and $p \not= 0$? How can we prove these formulas?
4
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1answer
98 views

Does $x^{x^{x^{x^{x^{\,\,\style{display: inline-block; transform: rotate(60deg)}{\vdots}}}}}} =y$ imply $2=4$? [duplicate]

Edit: My question has been requested to close due to its apparent lack of clarity. My question is below under "Problem". If the information above it is redundant, please let me know in a comment. I ...
0
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1answer
80 views

How to prove $(a \cdot b)^n = a^n \cdot b^n$?

Assuming $a, b \in \{\mathbb{R^-,R^+}\}$ and $n \in \mathbb{R}$. We often see the statement: $$(a \cdot b)^n = a^n \cdot b^n$$ Why we get the statement? How to prove this? There should pay ...
6
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0answers
141 views

Way to express a number in its most compact sum of powers

Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...
1
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1answer
61 views

Does this specific property exist for Modular Exponentiation?

$$a^{p-1} \equiv 1 \pmod p$$ Assuming the above expression is true, does it tell me anything about the congruence relation of the following expression: $$a^{\frac{p-1}{2}} \pmod p)$$ I can't seem ...
1
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1answer
74 views

Closed form for the series $a+a^p+(a^p)^p+\dots$

We have the arithmetic sequence in which each term is given by adding a common difference to the previous term. Then there is the geometric sequence in which each term is given by multiplying the ...
-2
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0answers
43 views

Power of Aleph 2

I would like to know if someone knows how to calculate $\aleph_{2}^{\aleph_{1}}$
0
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1answer
25 views

Power of prime ending with specified suffix

I've came across a problem I've got no clue how to tackle. Does such number exist, that it is a power of 7 and it ends with ...
0
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1answer
34 views

Solve to get a value for $y.$

Please help me solve this question. Solve for y. $1.\quad a^x=b^y=c^z$ $2.\quad b^2=ac$ I figured out that $b^{2+y}=a^{x+1}c^{z+1}$, but I am not able to go further, please help me.
4
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5answers
224 views

How to show that $\pi^\pi$ is not a integer?

This goes without saying, but, I can't use a calculator to evaluate $\pi^\pi$. I think we need to find a integer $x$ such that $$x<\pi^\pi < x+1. \tag{1}$$ However, since I have no ideia what $\...
2
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0answers
61 views

New inequality involving exponent

I'm interested by the following problem : Let $x,y,z>0$ then we have : $$x+y+z\geq \sqrt{\Big((x^xy^y)^{\frac{1}{x+y}}+(y^yz^z)^{\frac{1}{y+z}}+(z^zx^x)^{\frac{1}{z+x}}\Big)\Big(\sqrt{xy}+\...
3
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2answers
156 views

Which one is bigger $100^{300}$ or $300!$?

How to find which one is bigger $100^{300}$ or $300!$ without using a calculator? I have tried it for whole 2 years but could not find it yet.
0
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3answers
37 views

Prove $\forall k\geq 4,\log(1+x_k)-x_k\leq {-1\over6k}$ where $x_k={(-1)^k\over\sqrt k}$

The question: Prove $\forall k\geq 4,\log(1+x_k)-x_k\leq {-1\over6k}$ where $x_k={(-1)^k\over\sqrt k}$. The above inequality holds iff $$\begin{align} &\log(1+x_k)\leq x_k-{1\over 6k}\\ &\...
3
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2answers
246 views

Proof real to the power of real

Let a be a real number and p a non zero real number. Then $a^p$ is also a real number. What definitions, propositions and etc are required to prove this?
1
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6answers
193 views

Why does multiplication by purely imaginary numbers, regardless of magnitude, not cause purely rotation?

For context, I have been relearning a lot of math through the lovely website Brilliant.org. One of their sections covers complex numbers and tries to intuitively introduce Euler's Formula and complex ...
3
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2answers
118 views

Solving $\int ^{1}_{0}{(-1)^x}dx$

Does this integral actually have a value? I just wanted to check it's well-defined, since exponentiation gets complex when dealing with negative and complex numbers, as seen in post 1 and post 2. ...
2
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2answers
62 views

How is this read correctly (problem with brackets)? 2^2048 - 2^1984 - 1 + 2^64 * { [2^1918 pi] + 124476 }

How do you read this correctly, where are the brackets if you set them? 2^2048 - 2^1984 - 1 + 2^64 * { [2^1918 pi] + 124476 } Taken from: https://www.ietf.org/rfc/rfc3526.txt I have great problems ...
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1answer
83 views

Can you precisely calculate $x^{\pi}$ (up to first 200 decimals) without a computer

I have recently learned how to calculate $a^b$ for all $a,b \in \mathbb{Q}$. I have noticed that the $a$ in fraction form will always be $a\cdot 10^n/10^n$ where $n$ is $a$’s decimal length. But if $...
0
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0answers
54 views

Can any Power function (whose base is not zero) , e.g. $2^{n}$, be defined arithmetically (i.e. using addition and multiplication only)? [duplicate]

In other words, I'm looking for a binary relation $P(x,y)$, being arithmetical, i.e being expressed in the first order language of Peano arithmetic (hence non-recursively, i.e. using addition and ...
0
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0answers
25 views

help for tedious counting problem

for an integer $\mathscr k$ ($2\leq k\lt500$) four positive integer a,b,c,d satisfies following conditions $2\leq a,b,c,d\leq k$ $a^{\frac{1}{b}}\times c^{\frac{1}{d}}=24^{\frac{1}{5}}$ for k which ...
0
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1answer
46 views

Is there a method for computing logarithms similar to Square and Multiply for exponentiation?

I used the Square and Multiply method and now I was wondering if there is a similar thing for the computation of logarithms. Shouldn't it be exactly the opposite of Square and Multiply? And if so ...
2
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0answers
28 views

Exponents and order of execution [duplicate]

If presented the following, what order should the parentheses be added? 2^2^2^2 It seems most tools go right to left. Is that correct/expected? I could not find clear information on this. Thank you,...
0
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4answers
32 views

Removing expressions from exponentials

I have this expression: $$\exp(E_f/kt) = \exp(Ev_1/kt) / \exp(Ev_2/kt)$$ In the equation after this, all the $kt$ terms are removed and we are left with: $$\exp(E_f) = \exp(Ev_1) / \exp(Ev_2)$$ How is ...
0
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3answers
39 views

Estimating Area Exam Question

Pieces of turf are 1m long by 0.5m wide. Each piece costs £3.79 . $1 * 0.5 = 0.5\text m^2$ a)Estimate the cost of turf required to cover these spaces. i) 9.6m by 2.4m $10 * 2 = 20\text m^2$ ...
4
votes
2answers
207 views

Prove $0.9999^{\!101}<0.99<0.9999^{\!100}$

Prove $$0.9999^{\!101}<0.99<0.9999^{\!100}$$ I think its original idea is $$(1-x)^{(1-\frac{1}{x})}<(1+x)^{(\frac{1}{x})} \tag{I can't prove!}$$ For $x=100^{\!-1}\,\therefore\,(1-x)^{(1-\...
5
votes
3answers
101 views

Why do the factorials appear in differences of consecutive powers?

Why do the factorials appear when repeatedly taking the differences of consecutive powers? Or rather why is the $n_{th}$ factorial equal to the $n_{th}$ difference of $(k+1)^{n}-k^n$? I'm having ...
1
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8answers
75 views

Estimating Square roots

Question: Estimate the value to the nearest tenth $$\sqrt{47}$$ But I don't know how I could estimate without using the calculator Thank You and Help is appreciated
0
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2answers
51 views

isolating an exponent

It appears it's been a while since I've dealt with advanced algebra (or exponents outside of excel). I'm writing a tiny piece of software to solve for $n$. But I can't seem to isolate $n$ in the ...
4
votes
6answers
377 views

Approximate solution: factorial and exponentials

If z= $\dbinom{200}{100}/(4^{100})$, what is the value of z? The options are: a. $z<1/3$ b. $1/3<z<1/2$ c. $1/2<z<2/3$ d. $2/3<z<1$ How should I go about solving these type ...
1
vote
2answers
109 views

If $ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x $, what is the maximum $x$?

If $$ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x $$ Whati is the maximum value $x$ that fits in the equation? Attempt: $$ \sqrt{x}...
1
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2answers
52 views

Unintuitive behavior of x^y for x<0

I take advantage of mathematics a lot while programming, but in spite of that I'm very mathematically ignorant. Sorry if I'm asking something stupid. Often I need to exponentially amplify some values....