The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [exponentiation]

Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

Filter by
Sorted by
Tagged with
1
vote
2answers
56 views

Which is bigger 5^(√3) and 4^(√5)?

How do you know which one is bigger between $5^\sqrt{3}$ and $4^\sqrt{5}$? For my method I used in $2^\sqrt{3}$ and $3^\sqrt{2}$ I put both numbers in the function $f(x)=x^\sqrt{3}$ so $f(2^\sqrt{3})$...
0
votes
3answers
64 views

Exponential identity ${(x^a)}^b=x^{ab}$

We all know that $e^{\pi i}=\cos \pi + i \sin \pi=-1$, and that ${(x^a)}^b=x^{a \times b}$, also $e^{2 \pi i}=\cos {2\pi} + i \sin {2\pi}=1$. Here's my problem, we have $a \in \mathbb R$, and I ...
1
vote
0answers
53 views

Prime twins $ (3^n - 2, 3^n - 4) $ conjecture

Let $n$ be a positive integer. Conjecture There are infinitely many prime twins of the form $$ ( 3^n - 2, 3^n - 4) $$ Examples include $$(3^2 - 2,3^2 - 4) = ( 7,5 ) $$ $$ ( 3^{37} - 2 , 3^{37} - ...
-4
votes
0answers
24 views

Multiple term exponential inequality [on hold]

Show without a calculator that : $4^{79} < 2^{100} + 3^{100} < 4^{80} $ I would like to know how to solve the above problem without lengthy calculations. It is a university interview ...
1
vote
1answer
25 views

Expressing sums of complex exponential functions with no imaginary parts

I am trying to express this signal function as a sum of complex exponential signals $$s(t) = 10 + 20 \cos(200\pi t+\pi 4) + 10 \cos(500\pi t).$$ I know that $e^{ix} = \cos(x) + i\sin(x)$ and the ...
0
votes
1answer
38 views

insert an exponent of an integral into the integrated function

Suppose $x$ and $\lambda$ are real number, Are there any real-valued function $f(x,\lambda)$ and $g(x)$ satisfying following equation? $\int f(x,\lambda)dx=\left(\int g(x)dx\right)^\lambda$ General ...
2
votes
1answer
51 views

Is there any symbol for exponentiation series?

Summation series uses the Sigma symbol like this. $$\sum_{n=0}^{\infty} \frac{1}{2^n} = 2$$ Product Series uses the Pi symbol like this. $$\prod_{n=1}^{\infty} \frac{4n^2}{4n^2 - 1} = \frac{\pi}{2}$...
0
votes
1answer
23 views

Notation of a function raised to a power [duplicate]

$f(x)^2$ is sometimes written as $f^2(x)$. Why is it not written as $f(x)^2$? Is it just a matter of taste or convention?
1
vote
2answers
58 views

Solve $x,y,z, z^x=x, z^y=y, y^y=x$

I solved it and got three solutions $(1,1,1)$ is obvious, no need to calculate it. other two solutions are $(4,2\sqrt{2}),(4,2,-\sqrt{2})$ But actual answer is only $(1,1,1),(4,2,\sqrt{2})$. One ...
1
vote
2answers
92 views

How would I solve $53^{1069}$ mod 54? [duplicate]

I am doing practice problems for an upcoming exam and am wondering what approach I might take to solve $53^{1069}$ mod 54. 1069 is a prime number, which means I can't factor the exponent. Some other ...
1
vote
2answers
41 views

Why: $2^{n+1} = 1 + \sum_{i=0}^{n}2^i$ [duplicate]

I was doing some messing around on excel to answer this brain teaser: In autumn the amount of leaves falling from a tree get doubled after every hour. The tree is leafless after 9 days. How many ...
1
vote
3answers
118 views

How to rewrite $\lim_{h \to 0}\frac{e^h - 1}{h}=1$ into $\lim_{n \to +\infty}\left(\frac{x}{n}+1\right)^n=e^x$

I am trying to rewrite $\lim_{h \to 0}\frac{e^h - 1}{h}=1$ into $\lim_{n \to +\infty}\left(1 + \frac{x}{n}\right)^n=e^x$. Developing: $$\lim_{h \to 0}\frac{e^h - 1}{h}=1$$ $$\lim_{h \to 0}1 + h=\...
0
votes
1answer
48 views

How do $i$ and $\pi$ show up in this result from a Google search?

I am just curious and play a bit here. I googled (-1e-10)^(-1e-10) and got (-1e-10)^(-1e-10) = $1 - 3.14159266 × 10^{-10} i$ I guess this is some form of ...
2
votes
1answer
72 views

Left most Digits of $5^n$ [duplicate]

Prove that for every integer $m$ there is an integer $n$ such that the digits of $5^n$ start with $m$ (left most digits). For example for $m=156$, $n=6$ is the solution, because $5^6 = \color{red}{156}...
0
votes
1answer
35 views

Solve for positive integers $x$ and $y$

Find positive integers $x$ and $y$ such that:$$\frac{x}{y}=\frac{(x^2-y^2)^\frac{y}{x}+1}{(x^2-y^2)^\frac{y}{x}-1}.$$ I tried to proceed with this problem as follows: $$\frac{y}{x}=\frac{(x^2-y^2)^\...
0
votes
0answers
23 views

Equality of constant matrices' integration

Given that A and F matrices are constant n x n matrices show that: \begin{equation} e^{(A+F)t} -e^{At}=\int_{0}^{t}e^{A(t-\sigma)}Fe^{(A+F)\sigma}d\sigma\end{equation} I tried to group up the right ...
0
votes
0answers
40 views

Is there a name for a concatenation-exponentiation relation?

Suppose we define a set: $$ S := \{(x,y) \in \mathbb{N}\times \mathbb{N} | x^y =y \mathbin\Vert x\} $$ where "$\mathbin\Vert$" denotes the concatenation operation. We have the example of the pair ...
0
votes
0answers
35 views

Need help solving this system of equations

$$\begin{align} a+2.27 &= \ln(33a-14) * b(33a-15) * (33a-14)^c\\ c+1 &= bc(33a-15)(33a-14)^{c-1} + 33(33a-14)^c\\ d &= (33a-15)(33a-14)^c \end{align}$$ Solve for a in terms of d. Or just ...
1
vote
0answers
19 views

Why order matters in hyperoperations after exponentiation? [duplicate]

In first and second hyperoperations (addition and multiplication), order of two operands doesn't matter, result is same. However, it doesn't work on exponentiation. Why it does matter in ...
0
votes
0answers
12 views

Non-Assignment Question: Would like help finding cumulative # of hours spent in a 16-year period.

Hope you're doing well. PROPOSAL So, I’ve been trying to figure out how many hours of relief a clinic that I'm trying to help out has given to patients over a 16-year period. Each person treated by ...
2
votes
3answers
42 views

Simple exponents. Can't seem to solve this. $(2018-x)^{(221+x)}=1$

$(2018-x)^{(221+x)}=1$ Took log of both sides and then used power rule. I'm stuck now
-1
votes
0answers
41 views

Can someone explain how the approximation was made in this photo?

How did the last equation in short form come about from the previous equation? I am just not getting it. I even tried with the assumption given where epsilon << N/2 with and ended up with no ...
0
votes
0answers
18 views

Isolating variable that is an exponent and part of a modulus function

I am given $n$, $e$, and $c$, and I need to calculate the value of $ϕ(pq)$, however to do so, I need to isolate the variable from an equation. Currently, I found the following equations to use: $$c=...
2
votes
3answers
104 views

Compare numbers with big powers [duplicate]

Compare $2019^{2020}$ and $2020^{2019}$. I know that $2019^{2020}$ is greater than $2020^{2019}$ but I couldn't prove it. I tried proving $(\frac{2019}{2020})^{2019}.2019>1$ but without success.
1
vote
1answer
61 views

Proof check and hints?: $y=x^{1/n}\iff y^n=x$

I'm studying Tao's Analysis I, in which the reader has to prove many of the theorems and lemmas. $x$ and $y$ are nonnegative reals and $n$ is a positive integer. Definition of nth root: $x^{1/n}=\...
0
votes
1answer
40 views

Hyperoperations and Logarithms. [duplicate]

If Addition, Multiplication, Exponentiation are all iterated but they have been generalised(there is no problem even with a complex exponent) can Hyper(n) be defined for all n? Along with its hyper-...
1
vote
4answers
77 views

Numerical approximations of $2^x$ where $x$ is between $0$ and $1.0$?

I've been looking everywhere for this. I've found a billion ways to approximate e^x, and ln(x), and I know I could combine those, what I'm looking for here is an independent numerical method to ...
0
votes
0answers
18 views

Integrating exponentials (confused)

I am having some trouble with this integral term as I'm not sure whether to use the exponential integral Ei(x) = - $\int_0^\infty \frac{e^{-t}}{t}dt$ to integrate the following term $$\int_0^{25} e^{...
3
votes
2answers
62 views

A pattern on finding squares for which their sum is $\Big(\sum\limits_{i=0}^j x^i\Big)^2$

Apologies for the inactivity. I haven't been doing so well in life, lately, but I'm glad to be back at some maths! Here's a pattern I discovered. I'm not good at explaining with words, so hope you get ...
2
votes
1answer
43 views
0
votes
1answer
50 views

Does $(\omega^a)^b \neq \omega^{(ab)}$ for complex numbers $\omega, a, b \in \mathbb{C}$? [duplicate]

Im wondering wether the equality $$(\omega^a)^b = \omega^{(a\cdot b)}$$ hold for general complex numbers $\omega, a, b \in \mathbb{C}$? I tried some specific values and the result seems fine, but ...
3
votes
4answers
114 views

When is exponentiating both sides of equation an equivalent operation?

I have the following inequation: $\sqrt{t^{2}-t-12}<7-t$. Can I just set both sides of the inequation to the power of two, or is there any condition under which exponentiating is an equivalent ...
0
votes
1answer
56 views

An inequality about exponentials.

I have 2 questions, Is the following inequality true? If yes why? $$\frac{1}{2} \times ( e^{x} + e^{-x} ) \leq e^{\frac {x^2}{2}} $$ In general for real numbers $a, b$ do we have a nice ...
0
votes
0answers
44 views

Solving $M^2 = aN^p + yN^q$ for $N$

I have the following equation $$ M^2 = aN^p + yN^q\label{1}\tag{1} $$ where $a,p,y,q$ are constants, and I would like to express the above equation as a $$ N=f(M) $$ The dependence of $N$ on $M$ is ...
0
votes
0answers
31 views

Relationship of Power Law Scaling to Self-Similarity

Recently I was reading the book Scale by the theoretical physicist Geoffrey West. Much of the book is devoted how scaling relationships control the behavior of various phenomena, especially in the ...
-1
votes
1answer
86 views

How to derive $e^x = \lim_{n \to \infty} \left( 1 + \frac x n \right)^ n$ directly from $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^ n$? [duplicate]

If given the definition of $ e $: $$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^ n$$ Using this fact alone, can it directly derive $$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^...
2
votes
1answer
59 views

How can we prove for the solutions of the equation $x^y=y^z=z^a…$?

How can we prove for the solutions of the equation $x^y=y^z=z^a...$? You could natural log all of them and then proceed but I’m kind of confused. How would you solve for the solutions of a simpler ...
0
votes
0answers
31 views

Is it safe to assume that $x=y$ for the equation $x^y=y^x$? [duplicate]

This thought came to me when I was thinking about exponents. Is the only solution for $x^y=y^z$ be $x=y$? How exactly would we prove there are no other numbers or how would we exactly solve this?
1
vote
1answer
26 views

Exponentiating a 'polynomial' with non-negative real powers to produce a polynomial with integer powers

The following is something one of my teachers and I discussed but did not make any progress on, along with my own generalizations. Let $p(x)=x^\alpha$ with $\alpha \in \mathbb{R}^+$. If we let $q(x) =...
11
votes
4answers
460 views

Hard inequality :$\Big(\frac{1}{a^2+b^2}\Big)^2+\Big(\frac{1}{b^2+c^2}\Big)^2+\Big(\frac{1}{c^2+a^2}\Big)^2\geq \frac{3}{4}$

I have a hard problem this is it : Let $a,b,c>0$ such that $a^ab^bc^c=1$ then we have : $$\Big(\frac{1}{a^2+b^2}\Big)^2+\Big(\frac{1}{b^2+c^2}\Big)^2+\Big(\frac{1}{c^2+a^2}\Big)^2\geq \frac{3}{...
2
votes
1answer
56 views

Why does $n^{1/ln(n)} = e$?

As stated in the question, why does $n^{1/ln(n)} = e$?
2
votes
3answers
100 views

Comparing $a^b$ and $b^a$ when $b < e < a$

If $0 < b < e < a$, how can I determine whether $a^b$ or $b^a$ is greater? I know this question has been asked before, but I want to solve this question by this method. It worked fine for ...
0
votes
0answers
44 views

Why can't we define $0^0=1$, just like we did with $0!=1$? [duplicate]

$\bullet$ The $0!$ is something is pre-defined, and when even a calculator returns that it is because it is programmed to do so and doesn't actually do the calculation. It also cannot be expressed in ...
1
vote
0answers
58 views

Pattern in the differences between powers of 2 and 3

Let $n$ be some fixed, odd integer and $ q_k = 2^{\lceil \log_2(3^{k}n) \rceil} - 3^{k}n $ be the difference between $3^kn$ and the smallest power of two $\geq 3^kn$ . Now the following two things ...
2
votes
0answers
34 views

Well-definedness of rational exponents?

I am a little bit confused witht the definition of rational exponents. I believe that the definition of $a^{\frac pq}$ is $\sqrt[q]{a}^p.$ However, $\dfrac 26 = \dfrac 13$, yet $-1^{2/6}$ is undefined ...
1
vote
1answer
36 views

Exponent-factorial inequality marathon [closed]

A) I am wondering which one is bigger? $(((5!)!)!)!$ or $5^{5^{5^5}}$. B) And if there is a largest number with at most four $5$’s and four operations, or there is no such number. Here, we define ...
7
votes
1answer
103 views

What is the $n^\text{th}$ perfect power, $P(n)$?

Let $P(n)$ denotes the $n^\text{th}$ perfect power of natural numbers (in ascending order without repetition). So, $P(1)=1, P(2)=4, P(3)=8, P(4)=9, P(5)=16, P(6)=25, P(7)=27, P(8)=32, \dots$. Is ...
-1
votes
3answers
50 views

fractional powers of x [closed]

I would like a step by step explanation of the simplifications made to reach the result: $$\frac{30x^\frac{1}{2}y}{20x^{\frac{3}{2}}}=\frac{3y}{2x}$$
0
votes
2answers
57 views

The units digit of $(((\dots((2018^{2017})^{2016})^{.^{.^{.}}})^3)^2)^1$

I posted a problem, I got the answer from many guys, thanks for them. This is another problem, I am curious how to solve it. I tried to use modular arithmetic as in the problem linked above, but I ...
3
votes
0answers
93 views

$e^{i\theta}$ confusion [duplicate]

I learned the following $$ e^{ik2\pi}=1 $$ and I was wondering whether or not $k$ has to be an integer. Thought 1: Since $e^{ik2\pi}=\cos(2k\pi)+i\sin(2k\pi)=1$, equating the real and imaginary ...