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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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1answer
11 views

Finding the minimum number to be multiplied to reach the closest kth root

I was trying to solve this coding question. You are given an integer $N$ and a value $k$. You need to find a minimum number $X$ which when multiplied to $N$ results in the value of $N^{1/k}$ as an ...
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1answer
30 views

Induction on powers of powers.

So I would like to prove that $ a_n \; | \; a_{n+1} - 2 $, where $a_n = 6^{2^n} + 1 $ Now I know I need to do this by induction and so I begin by showing this is true for the base case. (Proposition)...
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0answers
46 views

Why isn't the first term of the McLaurin series for $\cos(x)$ a pole? [on hold]

As I understand it, the McLaurin series for $\cos(x)$ is $$\sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k)!}$$ This leaves me puzzled. Websites I've seen give the expansion of the sum as $$1-\frac{x^2}{...
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3answers
33 views

${x^4}$ as “tesseracting” a number $x$ [on hold]

So, this strange thought popped up into my head. You know how we call ${x^2}$ squaring due to the fact that what you're essentially doing is finding the area of a square with side length $x$? The same ...
3
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2answers
53 views

Tetration of non-integers: is there something wrong with this approach?

I'm trying to figure out a formula for tetration that will work for non-integer heights. I know the usual recurrence relation for tetration ($x \in \mathbb{R}, \text{ }n \in \mathbb{N})$: $${^{n}x} =...
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2answers
31 views

Trying to prove an equation

I would like to receive some help about the next problem. The problem: I'm trying to prove the next equation: $$\sum_{k = 0}^{n} \frac{(-1)^{-k}}{k!(n - k)!} = 0 \quad, n = 1, 2, ...$$ My work ...
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2answers
23 views

Smallest integer power for an inequality to hold

so I have this inequality: Given integers $m, k\geq1$. $$2^{m/k} > \frac{3}{2}$$ I'm interested in finding the smallest integer power $m$, as a function of $k$, that will make this inequality ...
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4answers
60 views

Comparing $\ln 1000$, $\sqrt[5]{1000}$, $3^{1000}$, and $1000^{15}$ without calculator

In my Pre-Calculus class we were given the following problem: Put the following four values in order from smallest to largest: $\ln 1000$, principal $5$th root of $1000$, $3^{1000}$, and $1000^{15}$...
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1answer
20 views

Proof of exponentiation law

I want to prove this : $(ab)^n = a^nb^n$ with a, b and n real numbers. I know how to prove this when n is an integer but not when n is a real number. I really don't know where to start to prove this. ...
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5answers
598 views

Proving that $\lim_{n\to\infty} \left(1+\frac{1}{f(n)}\right)^{g(n)} = 1$

I want to prove that $$\lim_{n\to\infty} \left(1+\frac{1}{f(n)}\right)^{g(n)} = 1$$ if $f(n)$ grows faster than $g(n)$ for $n\to\infty$ and $\lim_{n\to\infty} f(n) = +\infty = \lim_{n\to\infty}g(n)$. ...
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1answer
21 views

Whether a Pathological Set Might Exist that could Foil a Theorem about the Exponential Function

There has been some discussion recently about the theorem $$\lim_{n \rightarrow \infty} \prod_{i = 1}^n \left( 1 + \frac{x_i}{n} \right) = \exp \left( \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i =...
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0answers
12 views

Can raising a number to an irrational power have infinite solutions?

$a^{\frac{1}{2}}$ is generally considered to be the positive square root of $a$, but it also makes sense (depending on context) to consider it to be multivalued, returning all square roots of $a$ ...
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2answers
57 views

Which of the following has the greatest value

Which of the following has the greatest value? a) $2^{64}$ b) $4^{63}$ c) $8^{34}$ d) $16^{17}$ I tried finding a pattern among exponents and their is none. but there is a pattern in base, but I'...
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1answer
26 views

What to do if a number becomes bigger than 10 on scientific notation when performing arithmetics with it?

For example, I was adding (9.99 x 104)+ (9.99 x 105). I raised the first number's exponent from 4 to 5 by dividing it by 10 once more, thus getting 0.999 and the 105 exponent as a way to reverse it (...
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6answers
99 views

The expression: $-5^2$

A few people I met were debating over if $-5^2 = 25$ or $-25$. From my experience, we assume operator precedence and get $-25.$ People are telling me however, calculators are flawed, the real answer ...
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3answers
34 views

Basic Mathematics, Rules for Multiplication trouble

Doing some self study from the text Basic Mathematics by Serge Lang I ran into an exercise question which I can't seem to wrap my head around. The question is: Express the following expressions in ...
2
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1answer
24 views

Representing positive integers as floor of integer powers of real number.

Does there exist real $a$ such that for every positive integer $c$ there exists integer $b$ such that $\lfloor{a^b}\rfloor = c$?
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2answers
35 views

Exponent laws for rational numbers

this seems to be easy but I dont manage to prove it properly. You are only allowed to use the following laws: $$\left(z^m\right)^n=z^{m\cdot n}\quad \text{for $n,m\in \mathbb N$}$$ and $$z=a^{\frac ...
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2answers
27 views

Larger value with right associative tetration?

Given right associative tetration where: $^{m}n =$ n^(n^(n^…)) And a situation such as: $^{m}n = y$ $^{q}p = z$ What is a practical way to calculate which of $y$ and $z$ are larger? I'm ...
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2answers
37 views

Why are negative exponents dividing instead of multiplying?

In trying to relearn scientific notation after years since I left school, I noticed that when we get a very small number and convert it to scientific notation, you're actually multiplying the tiny ...
2
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2answers
113 views

Why are we using “Euler's Number” constantly? [duplicate]

I am a student at technical university.We are currently studying Calculus and I am really curious about why we are using Euler's Number.
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2answers
74 views

Proof verification: rational solutions to $x^y=y^x$

I am trying to find rational solutions to the equation $x^y=y^x$, but I'm unsure if my solution is right (especially the first paragraph below). Here is my attempted solution (I've excluded the ...
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3answers
78 views

Proof that the square root of a negative number is real.

So I stumbled upon this weird result when experimenting with fractional exponents. Suppose you have some negative, real number, for example -8. We know $\sqrt{-8}$ is not a real number.But $$\sqrt{-8} ...
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2answers
20 views

Does $| E[X*1\{X<\mu\}] | = | E[X*1\{X>\mu\}] |$?

Does mean of a random variable $\mu=E[X]$ devide conditional mean in half, in a sense that $$ | E]X*1\{X<\mu\}] | = | E[X*1\{X>\mu\}] |$$ or is this only true for symmetric distributions?
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0answers
16 views

numerical analysis and weakness of calculator

Suppose that our calculator can calculate $e^x$ so well.But its program for calculating $\ln (x)$ is poor. How to improve the accuracy of $ln(x)$ by using the fact that $$ln(a)=b+ln(1+\frac{a-e^b}{e^...
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1answer
32 views

Solving modular exponentiation

Calculate : $(8^{2^{6^{4^{2^{5^{8^9}}}}}}) (\mod 10000)$ But, the problem is that $8$ and $10000$ are not co-prime. Moreover, the goal is to use Euler's theorem (modified?) to solve this. Any help is ...
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0answers
19 views

Function mapping a number $\mod 8$ and power of 2 representation

Define $n\in\mathbb{N}$ Let me also represent the number $n$ in exponential form as the sums of power of twos. How can we then remove the powers where the exponent is greater than $2$? So what we ...
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0answers
18 views

exponents and iota

lets look at the below derivation, $(x)^\frac{a}{b} = (x^a)^\frac{1}{b}$ squaring both sides, $(x^\frac{a}{b})^2 = ((x^a)^\frac{1}{b})^2$ $(x^\frac{a}{b})^2 - ((x^a)^\frac{1}{b})^2 = 0$ $(x^\frac{a}...
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1answer
36 views

Complex numbers, finding smallest exponent

$$ z= \frac{\sqrt 3 -i}{1+ \sqrt3 i}$$ I need to find smallest exponent $n>2018$, such as $z^n$ will be a number with real part equal to $0$ and imaginary part of number will be negative. I ...
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2answers
44 views

Equivalence of all representations of $\exp$

I can name at least 4 different ways of representing $\exp$ function: Taylor series: For $x \in \mathbb{R}, \exp(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!}$. Differential equation: $f: \mathbb{R} \to \...
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1answer
54 views

What's equal the below power nested radical?

it is well known that $$\frac{2}{\pi}=\sqrt{\frac12}{\sqrt{\frac12+\frac12\sqrt{\frac12}}{\sqrt{\frac12+\frac12\sqrt{\frac12+\frac12\sqrt{\frac12}}}{\sqrt{\frac12+\frac12\sqrt{\frac12\cdots}}}}}$$ ...
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1answer
40 views

Find the limit of complex number exponentiation

How do I find the limit of $$ Z_n=\left(1+{\frac{a+bi}{n}}\right)^n $$ Should I have real and imaginary parts? Like this $$ \lim Z_n = \lim \left(\left(1+{\frac{a}{n}}\right)+{\frac{b}{n}}\right)^n ...
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1answer
40 views

Is the 4th root of $3^3$ $3^3/4$ or $2.2795$?

I'm working through a textbook and one question is: Use a calculator to find the value of the following expression: $$\sqrt[\large4]{3^3}$$ The textbook answer is given as $2.2795$; however, using ...
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2answers
20 views

Find all values of given degrees

I have $1^i$. To find all values, I do $$1^i=e^{\operatorname{Ln}1^i}=e^{i\operatorname{Ln}1}$$ Since $\operatorname{Ln}Z=\ln|Z| + i(\operatorname{arg}Z + 2\pi k)$, therefore $\operatorname{Ln}1=2\pi ...
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1answer
45 views

If matrix $X$ is sparse, does it mean that $e^X$ is also sparse?

Let's say, that we have a sparse matrix $X$. Then, is $e^X$ also sparse? At first glance I would say, that it is not always true, because we could expand this function in Taylor series and I think, ...
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3answers
28 views

Computing 2 to the power of some value without calculator

So i have an upcoming exam, and since no calculators is allowed, i was wondering if there is an approach to calculating the value of 2 to the power of some value? For example, ...
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0answers
25 views

Limit of infinity to the power of infinity vs factorial

I've been playing around with Taylor polynomials for sin(x) and it makes sense that the polynomial series converges to sin(x): For x as a reasonable number, the ...
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2answers
57 views

When does $a^x = k\cdot x^b$ have a unique “nice” solution?

I'm looking for an example of a transcendental equation with a unique "nice" solution that can be identified as correct by inspection. My first thought was $$3^x = x^3$$ but this equation has two ...
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0answers
66 views

Why is $x=4$ as a fixed point of a map $\sqrt{2}^{x}$ unstable?

My question is motivated by this What is wrong with this funny proof that 2 = 4 using infinite exponentiation? discussion, namely an example of a map $x \mapsto f(x)$ is given, with $$f(x)= (\sqrt{2})^...
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2answers
63 views

Chinese remainder theorem to find $1030^{989}\bmod\; 3003$?

so this is a slightly different take on a question I asked, but instead of the product of two numbers- this time it is a very large number raised to a very large power. I am meant to use the Chinese ...
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0answers
36 views

I don't know how to approach a limit problem.

I need to find the limit of $$ \lim_{n \to \infty} \Big(\frac{1+n^2+n^3+...+n^{10}}{2+n^2+n^3+...+n^{10}}\Big)^\sqrt[n]{4}. $$ First of all I found the limit of $\sqrt[n]{4}$ is $1$. Now I don't know ...
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1answer
60 views

Number in power of $ log x$ stuck

Find $x$ if $$x^{\log 26} - x^{\log 24} = x$$ I can’t fiqure this out , I read all law of log but unfurtunately I can’t solve this. Any help about this? Sorry for my bad english.
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0answers
14 views

Prove that if $a$ and $b$ are coprime and the product $ab$ is some $m$-th power ($m\ge2$), then $a$ and $b$ have to be $m$-th powers. [duplicate]

i.e $ab=c^m$ for some $c\in\mathbb{Z}$, $m\in\mathbb{N}$. So far I have Let $p_1,\dots,p_n,q_1,\dots, q_k$ be primes $$a=p_1^{x_1}\cdot p_2^{x_2}\cdot ... \cdot p_n^{x_n}$$ $$b=q_1^{y_1}\cdot ...
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0answers
20 views

Montgomery Reduction - what should the choice of R be?

When we need to compute $z = xy \text{ mod } N$ and the Montgomery Reduction of $x$ is $xR^{-1}$ why should the choice of R be $2^l$ where $l$ is the length of $N$ to the base $2$? Why cannot we have ...
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1answer
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Simplifying equations while holding some constraints

I have two equation and i tried to solve them but i can't simplify them : $(\frac{1}{b})^{\frac{1}{r}} = 0.8$ $ br = n$ $r = \frac{n}{b} \rightarrow \frac{r}{n} = \frac{1}{b}$. Plugging this ...
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1answer
36 views

Applying rules of algebra when working with multiplication and exponents

I'm taking an online course and help is hard to find. This specific problem has to do with recurrence relation. I apologize for being too general but I'm just looking for help in how to go about ...
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2answers
117 views

Solve for $x$ given $2.544 = e^{-x} + e^{-2x} + e^{-3x}$ [closed]

Can this problem be solved algebraically? $$e^{-x}+e^{-2x} + e^{-3x} = 2.544$$
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2answers
71 views

Showing that $2^{2^{2^x}}<100^{100^x}$ for large $x$

I wish to show that $2^{2^{2^x}}<100^{100^x}$ for $x$ sufficiently large. I have taken logs (base 10) of both sides to get $2^{(2^x-1)}\log_{10} 2$ and $100^x$. It is not immediately clear how I ...
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2answers
29 views

how to evaluate a exponent raised to a log of same base?

why is $2^{log_2(n-1) + 1} = 2(n-1)?$ I tried using the formula $b^{log_b(n)} = log_b(b^n) = n$ $\,$ but can't seem to get $2(n-1)$ please help
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1answer
37 views

Can we have a cardinal exponent of any number?

Can we have a cardinal such as ${1.5}^{\aleph_0}$ in the way that we have a powerset as $2^{\aleph_0}$, or is $2^{\aleph_0}$ just the notation we use, rather than actual exponentiation. Or is ...