Questions about exponentiation

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3
votes
3answers
138 views

$2^{10 - x} \cdot 2^{10 - x} = 4^{10-x}$

$$2^{10 - x} \cdot 2^{10 - x} = 4^{10-x}$$ Is that correct? I would've done $$ 2^{10 - x} \cdot 2^{10 - x}\;\; = \;\; (2)^{10 - x + 10 - x} \; = \; (2)^{2 \cdot (10 - x)} \;=\; 4^{10 - x}\tag{1} $$ ...
1
vote
2answers
99 views

Understanding $\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$

I'm going through a book about algorithms and I encounter this. $$\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$$ How is this equation formed? If a theorem has been applied, what theorem ...
8
votes
1answer
150 views

Proving a number defined by a sequence is a square number

I found this problem in a math magazine: Given the sequence $(x_n)_{n \in \mathbb{N}}$ defined by: $$ x_0 = 0\\ x_1 = 1\\ x_{n+2}+x_{n+1}+2x_{n}=0 $$ Prove that $s_n = 2^{n+1}-7x_{n-1}^2, n ...
22
votes
10answers
4k views

How to solve $x^{1/2}-x^{1/3} = 0$

How can I solve the following equation? I really can't figure out how to solve it: $x^{1/2}-x^{1/3} = 0$ Thank you.
0
votes
1answer
18 views

How to find an algebraic term when 2 similar exponential forms of it are given?

if $a^2bc^3 = 25$ and $ab^2= 5$, how would you find $abc$? Is there a formula of which these are a part of? What I've done is this: $$abc^3 = \frac{25}a\quad\implies\quad abc = ...
6
votes
1answer
503 views

System of Equations With Exponents

What are the steps to solving a system of equations when $x$ and $y$ are exponents? Here is the problem to solve for $x$ and $y$. $$8^{3x}=4^{2y}$$ $$x-y=5$$
1
vote
5answers
128 views

Solution for $\log_7x+\log_{\frac17}x^2=\log_{49}x-3$

What is the right solution for $\log_7x+\log_{\frac17}x^2=\log_{49}x-3$. What logarithm identities used?
1
vote
0answers
77 views

Use of natural logarithm transformation on weighted index series

I have a value computed as sum of powers, e.g. $x^5+y^8+z^2$. The exponent represents the weight for variables, $x, y$ and $z$ in the example above. Applying natural logarithm on $x^5+y^8+z^2$, I get ...
2
votes
2answers
81 views

Proving that $x^a=x^{a\,\bmod\,{\phi(m)}} \pmod m$

i want to prove $x^a \equiv x^{a\,\bmod\,8} \pmod{15}$.....(1) my logic: here, since $\mathrm{gcd}(x,15)=1$, and $15$ has prime factors $3$ and $5$ (given) we can apply Euler's theorem. we know that ...
1
vote
1answer
114 views
0
votes
2answers
78 views

What are these numbers called?

Say I have numbers that are all multiple of 2, I would say, well they are multiples of two. How are numbers $x$ called with respect to $a$ that are all formed like $x = a^b$? I am assuming here that ...
1
vote
4answers
664 views

Value of $k$ for which $e^x = kx$ has $1$ solution

I need to work out the value of $k$, where $k>0$, for which $e^x=kx$ has $1$ solution. I've done it somewhat intuitively as follows: $e^x=kx$ By inspection we can see that when $x=1$, the ...
6
votes
3answers
186 views

What's the intuition behind non-integer exponents/powers

Consider some $a \in \mathbb{R}$ and $x \in \mathbb{R}\backslash \mathbb{N}$. Is there some intuition to be had for the number $a^x$? For example the intuition of $a^2$ is obvious; it's $a*a$ which ...
4
votes
3answers
163 views

What is the difference between exponentials and powers?

I am a java programmer. But I have a doubt regarding a mathematics. There was a method called Math.exp(double a) description:Returns Euler's number e raised to the power of a double value. and another ...
2
votes
5answers
277 views

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k$?

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k\;$? I'm aware that $3 = 3^1$ but I would expect $3\cdot 3^k\;$ to be $\;9^k$ or $\;9^{k+1}$.
4
votes
1answer
95 views

What's the arity of the factorial and exponential operations?

I'm having a conflict with the concept of arity, I've read that the factorial is a unary operation and also that the exponentiation is a binary operation but I feel there's something strange, the ...
5
votes
0answers
219 views

Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} ...
4
votes
1answer
147 views

Operators - sums, products, exponents, etc.

$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$. $(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$. Is there an operator, such that if ...
1
vote
1answer
143 views

Solve for variable inside multiple power in terms of the powers.

I'm a programmer working to write test software. Currently estimates the values it needs with by testing with a brute force algorithm. I'm trying to improve the math behind the software so that I can ...
0
votes
0answers
43 views

can´t solve this kind of exponential equations

I have no clue how can i solve this kind of exponential equation in closed form: $a^n - b^n \le c$ where $a > 1$ and $-1 < b < 0$ thank you very much for your help
0
votes
2answers
73 views

Computing $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$

I'm trying to find $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$ mod $m$. $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\varphi^3 = 2 + \sqrt{5}$. But honestly I'm not even sure where to start. ...
2
votes
0answers
195 views

Poisson exponentiation distribution family and convolution

Assume $\xi_i \sim \mathbb{F}_{\lambda_i}(x)$ are random variables from Poisson distribution. Consider random variables $\eta_i \sim \tilde{F}_{\lambda_i,t}(x)$, where $\tilde{F}_{\lambda_i,t}(x) = ...
9
votes
5answers
471 views

Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}$

I'm so puzzled about this: $$a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}.$$ Why isn't $a^{b^c}$ equal to $a^{(bc)}$? Why is $a^{b^c}$ instead equal to $a^{(b^c)}$? And how is it possible that ...
1
vote
0answers
172 views

Power sums, fast algorithm

I know some schemes to compute power sums (I mean $1^k + 2^k + ... + n^k$) (here I assume that every integer multiplication can be done in $O(1)$ time for simplicity): one using just fast algorithm ...
11
votes
4answers
711 views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
20
votes
18answers
2k views

How to understand why $x^0 = 1$, where $x$ is any real number?

Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$ I understand that the ...
4
votes
1answer
209 views

Summing 2 to the power of the subset sums of a power set

Sorry in advance, as I suspect I lack both the proper terms and the proper notation for the problem I have, but I'll try to be clear. If I have a set $S = \{1,2,3\}$, I figured out that the summation ...
3
votes
5answers
169 views

Summation over exponent $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$

Why does $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$, where does that 3 comes from? Ok, from your answers I looked it up on wikipedia Geometric Progression, but to derive the formula it says to multiply by ...
4
votes
7answers
587 views

Numbers to the Power of Zero

I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations. This is a three part question, the parts are as ...
1
vote
0answers
89 views

Can exponentiation and power function be defined through Albert Bennett's operations?

In 1914 Albert Bennett suggested the following operation: $$a * b=a^0_2b=\exp(\ln a \ln b)$$ Now, given this function, addition and multiplication, and their properties, can one express ...
0
votes
1answer
107 views

How do I solve this exponential equation?

$$x = 2^{x-3}$$ Does there exist an analytical solution to this equation? If so, how do I find it? What if it is changed to an equality? $$x>2^{x-3}$$
1
vote
4answers
880 views

Integration with infinity and exponential

How is $$\lim_{T\to\infty}\frac{1}T\int_{-T/2}^{T/2}e^{-2at}dt=\infty\;?$$ however my answer comes zero because putting limit in the expression, we get: $$\frac1\infty\left(-\frac1{2a}\right) ...
0
votes
1answer
110 views

Solution for an exponential expression without using logarithms, with two defined variables

If $60^a=3$ and $60^b=5$, what is the result of $12^{\frac{1-a-b}{-2-2b}}$? This has to be done without logarithms. The past four hours were helpless to me. Any hint, solution is welcome, I just ...
3
votes
1answer
155 views

Diophantine equation: fermat numbers and fibonacci numbers

My question is how to find all solutions $(m,n)\in\mathbb N^2$ for $F_n=f_m$, where $F_n=2^{2^n}+1$ and $f_m$ is the $m$th fibonacci number: $f_0=0$, $f_1=1$ and $f_n+f_{n+1}=f_{n+2}$ for each ...
1
vote
2answers
621 views

Solving system of recurrence relations

Base Case: $$ \left\{ \begin{array}{c} T(1) = 1 \\ T(2) = 1 \\T(3) = 4\end{array} \right. $$ I have the system: $$ \left\{ \begin{array}{c} T(N) = G(N-1) + F(N-1) \\ G(N) = F(N-1) + G(N-1) \\ ...
5
votes
7answers
464 views

Is $0^0=1$ postulate independent of all other axioms of complex numbers?

This question is inspired by the other question which asked for a proof that $i^i$ is a real number. Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
0
votes
2answers
1k views

Find the remainder of $128^{1000}/153$.

I have tried for almost more than an hour to find the remainder of the following $$\frac{128^{1000}}{153}$$ I applied remainder theorem to get the answer but could not succeed. Any suggestion or ...
2
votes
2answers
186 views

Find the greatest powers of $2$ dividing $10!$, $20!$, $30!$, $40!$ [duplicate]

I'm trying to find the greatest powers of $2$ dividing $10!$, $20!$, $30!$, $40!$, as part of a basic number systems course. I'm rather lost with this question. For $10!$ I tried writing the terms ...
7
votes
1answer
709 views

Why use radical notation instead of rational exponents?

I'm helping my younger sister for her math class. She has recently been taught integer exponents, and has starteed studying radicals (mainly square roots). The next topic will be rational exponents, ...
0
votes
2answers
77 views

Exponentation vs Power

What definition of $a^b$ operation is the most popular and standartized: Exponentation or Power? Is any difference between them?
4
votes
3answers
183 views

Solve $3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$

$$3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$$ I am completely lost on how to proceed. Could someone explain how to find any real solution to the above equation?
4
votes
2answers
627 views

upper bound of exponential function

I am looking for a tight upper bound of exponential function (or sum of exponential functions): $e^x<f(x)$ when $x<0$ or $\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$ Thanks a ...
1
vote
6answers
721 views

Identity proof $(x^{n}-y^{n})/(x-y) = \sum_{k=1}^{n} x^{n-k}y^{k-1}$

In a proof from a textbook they use the following identity (without proof): $(x^{n}-y^{n})/(x-y) = \sum_{k=1}^{n} x^{n-k}y^{k-1}$ Is there an easy way to prove the above? I suppose maybe an ...
8
votes
2answers
322 views

Is there another representation for $x^x$

I started wondering about this the other day. Since the following have their own alternate representations. $$\begin{align*} \displaystyle\large x+x=2x & \ \frac{x}{x}=1 & xx=x^2\end{align*}$$ ...
1
vote
1answer
53 views

Conventions for notation of function exponentation.

I read a previous question here but it seems incomplete for me (missing references). Given a generic function, $ f $ : 1. is true that $ f^2 $ means $ f^2(x) = (f \circ f)(x) = f(f(x)) $ ? 2. or is ...
1
vote
0answers
121 views

Solving for $x$ in $y=x^x(\ln x + 1)$ (Lambert W?)

I made a bunch of problems exercising the Lambert W-function in the solution, because I like to exercise to new concepts that I learn about. One that I came up with was rearranging $y = x^x(\ln x + ...
3
votes
1answer
194 views

Summation of powers inequality

Can anyone provide a slick proof of the following? Let $0 < x \le 1$. Then $\displaystyle \sum_{k=0}^{n-1} x^k \ge \frac {1} {1 - (1 - 1/n)x}$.
2
votes
0answers
435 views

exp(ab) decomposition

How can one write $e^{a(x) \cdot b(x)} = c(x) e^{ b(x) }$ with $c(x)$ not implicitly depending on $b(x)$. I do not believe this is generally possible so alternatively one can use an infinite series or ...
2
votes
2answers
58 views

regarding exponents, how to interpret and use.

Some times in the books both of the below mentioned concepts are used interchangeably. Is there any reason for that? When to use -(2)^2 = -4 and (-2)^2. Explain with any useful examples.
36
votes
11answers
3k views

How is $e^x$ read aloud?

My current research colleague from New Castle told me that I was reading it wrong. I usually read it as e power x. How do you read aloud $e ^ x$? Is it: e raised to x e power x e powered x or e ...