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

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0
votes
2answers
89 views

Converting 29^1312000 to base 10

I am trying to do some calculations with the number 291312000 and I find it would be much easier if I could convert it (approximately) to a base 10 number. The closest I could come was to start with ...
3
votes
6answers
120 views

How to evaluate $\lim \limits_{n\rightarrow \infty}(n+9)^{\frac{1}{n}}$?

I have no idea where to begin to evaluate $\lim \limits_{n\rightarrow \infty}(n+9)^{\frac{1}{n}}$. The given answer is $1$. I do know that $\lim \limits_{n\rightarrow \infty} n^{\frac{1}{n}}=1$ (my ...
2
votes
2answers
54 views

A limit question involving power of positive numbers

I'm trying compute the following limit: $$\lim_{t\to0}\left(\frac{1}{t+1}\cdot\frac{b^{t+1}-a^{t+1}}{b-a}\right)^{1/t},\quad b>a>0.$$ I know $\displaystyle\lim_{t\to0}(1+t)^{1/t}=e$ and ...
32
votes
10answers
3k views

If both $a,b>0$, then $a^ab^b \ge a^bb^a$

Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
4
votes
2answers
460 views

Solving an equation with exponentials

$$2^x+4^x+12=0$$ How exactly am I supposed to solve this? Am I supposed to get $x$ alone or solve it another way?
1
vote
1answer
85 views

Finite natural summation that leads to double exponential results

We know that $$f(n)=\sum_{i=0}^n\binom{n}{i}=2^n$$ and $$g(n)=\sum_{i=0}^ni\binom{n}{i}=n2^{n-1}.$$ Are there any finite natural sums that lead to $2^{2^n}$ or $2^n2^{2^{n-1}}$ results other than ...
0
votes
1answer
33 views

Combining two fractions involing powers of x

Is there any way i can write $x^a+x^b$ as d$x^c$ Im considering writing letting $a=a-1$ and partial fractions but im getting really confused.
4
votes
1answer
124 views

asymptotic of $x^{x^x} = n$

How find the asymptotic behavior for $x(n)$ if $x^{x^x} = n$? I supposed that $x = O(\log\log{n})$ and took logarithm two times. So I get $x = O(\frac{\log\log{n}}{\log\log\log{n}})$ Is it right? ...
0
votes
1answer
54 views

Variables, Square roots, and exponents

Answer : $x^2$ I got $x^n$, shouldn't I be multiplying the variables in the parentheses first. Thus cancelling out the roots and left with $x$ then to the power of n? thus -> $x^n$ ? Please explain ...
0
votes
2answers
46 views

Simplifying exponents (How does $(2^6 + 2^3)(2^6-2^3)$ get created from $(2^6)^2 - (2^3)^2$

How does $(2^6 + 2^3)(2^6-2^3)$ get created from $(2^6)^2 - (2^3)^2$? Can someone explain?
0
votes
1answer
13 views

Powers inequality proof

I don't even understand what this proof is asking, let alone how to do it. here it is: Show that if $x>1$ is a real number and if $a<b$ are rational numbers, then $0\le x^a \le x^b$. any hints ...
0
votes
1answer
56 views

if -a^(-b^-c) is a positive integer and a, b, and c are integers, then…

(a) a must be negative (b) b must be negative (c) c must be negative (d) b must be an even positive integer (e) none of the above
1
vote
3answers
46 views

How to prove that $5^q<7^q$ implies $q>0$?

Consider $q\in \mathbb{Q}\:$ , $5^{q\:}<\:7^q$ How to prove that $q>0$ using only Power rules? So far, I just know that if $q=0$ we get $1>1$ and its false. But what to do get contradict ...
1
vote
4answers
75 views

Why can't you multiply the exponents when you have addition involved?

For example, $(8^{\frac{1}{3}} + 27^{\frac{1}{3}})^{2}$ why can't you make this $8^{\frac{2}{3}} + 27^{\frac{2}{3}}$? Please explain in a very simple way, thank you :)
2
votes
3answers
132 views

Why is x^(a/b) equivalent to the bth root of x raised to the a power? [duplicate]

I was wondering if someone can tell me what the logic behind converting fractional exponents to radicals is? For example, the exponent 1/2 is a square root, 1/3 is a cube root, and 2/3 is the cube ...
1
vote
2answers
59 views

Raised to the power and modulus

Task: $26^{61}(\pmod {851}$ And I stucked with the operation pow(26,61) because it's too hard for me. I read the article about this problem, but I don't quite understand how to solve it. I can ...
7
votes
1answer
157 views

Number theory: $x^y + 1 = y^x$

Today a friend told me the equality: $2^3 + 1 = 3^2$, and I wondered if there exist more solutions to the general problem $$x^y + 1 = y^x$$ where $x$ and $y$ are integers. Some research led me to the ...
1
vote
1answer
95 views

Conjecture: three or more decompositions into powers with a base differing by 1 means its a perfect power

If $$(i_1)^{a_1}(i_1+1)^{b_1}=n $$ $$(i_2)^{a_2}(i_2+1)^{b_2}=n $$ $$(i_3)^{a_3}(i_3+1)^{b_3}=n $$ where all the terms are positive integers and the groups ...
0
votes
3answers
61 views

Moving an exponent from the top to the bottom of a fraction and vice versa? Help pretty please :)

So I know that $x^{-1} = 1/x$ by definition. yeah okay, why can't you move a variable with an exponent to the top or bottom of a fraction when you have addition or subtraction involved? for example. ...
18
votes
9answers
2k views

How to determine without calculator which is bigger, $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ or $\left(\frac{1}{3}\right)^{\frac{1}{2}}$

How can you determine which one of these numbers is bigger (without calculating): $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
8
votes
2answers
233 views

What is the function $f(x)=x^x$ called? How do you integrate it?

For real numbers $x > 0$, the function $f(x)=x^x$ seems pretty cool. Is there a name for this function? It's obviously been studied before. It grows faster than exponential functions and ...
2
votes
2answers
49 views

Can every positive integer be expressed as a difference between integer powers?

In mathematical notation, I am asking if the following statement holds: $$\forall\,n>0,\,\,\exists\,a,b,x,y>1\,\,\,\,\text{ such that }\,\,\,\,n=a^x-b^y$$ A few examples: $1=9-8=3^2-2^3$ ...
1
vote
2answers
102 views

Using the basic laws of exponent [closed]

I have some problems with this question. Please help me. Thanks Simplify given expression$$ a^2 (abc)^{-2} a^3 b^7 $$ What are exponents of $a$, $b$, and $c$? I get $3,5,-2$ as exponents of ...
1
vote
2answers
72 views

Is there analytical solution to $x^{a+1} + x^{a} = c$ with $0<a<1$?

How to solve the nonlinear equation of type $x^{a+1} + x^{a} = c$ with $0<a<1$ and $c>0$? Sorry, I don't know which tag is appropriate.
0
votes
2answers
49 views

Is the sum of rational exponentials a rational exponential?.

Prove or disprove that $\forall a,b \in \mathbb{Q}^+$ and $ \forall p,q \in \mathbb{Q}$ there exists $c \in \mathbb{Q}^+$ and $r \in \mathbb{Q}$ such that: $$ a^p+b^q=c^r $$
5
votes
2answers
185 views

Ordinal exponentiation identity with natural sum of exponents

This is related to a previous question on How to think about ordinal exponentiation? One possible definition for the natural product $\alpha\otimes\beta$ of ordinals is based on Cantor Normal Forms ...
0
votes
1answer
20 views

$e^z=-3i$ find $z\in \mathbb C$ check my answer

I am unsure of my solution to this question, since the definition of the complex logarithm is somewhat complex. Since $-3i = 3e^{i\frac{3}{2}\pi}$ we get that $e^z=3e^{i\frac{3}{2}\pi}$ So if we use ...
2
votes
2answers
38 views

Arranging set A and B to maximize their power

Given two sets A and B each with $n$ positive reals. How to arrange elements in A and B such that $$\prod_{i=1}^n a_i^{b_i}$$ is maximized? Will ascending order of A and B make the correct ...
5
votes
2answers
119 views

$(-32)^{\frac{2}{10}}\neq(-32)^{\frac{1}{5}}$?

Is exponentiation by rational numbers defined only for simple fractions? $(-32)^{\frac{2}{10}}=\sqrt[10]{(-32)^2}=\sqrt[10]{1024}=\pm2$ (and $8$ other complex roots) ...
0
votes
4answers
155 views

Why is $0^0$ also known as indeterminate? [duplicate]

I've seen on Maths Is Fun that $0^0$ is also know as indeterminate. Seriously, when I wanted to see the value for $0^0$, it just told me it's indeterminate, but when I entered this into the exponent ...
3
votes
3answers
193 views

$1^x = 1^y$ and $x,y$ belongs to Real Numbers.

$1^x = 1^y$, and $x,y \in \mathbb{R}$. Following the rule, same base has powers equal every $x$ should be equal to every $y$. $$1^x = 1^y$$ $$x = y$$ What went wrong?
-2
votes
1answer
51 views

Is this identity correct?

Is this identity true? Wolfram|Alpha thinks is not. $$x^{ln(x^3)} = e^{3\,[ln(x)]^2}$$ That's how I demonstrated it: $${\left(e^{ln(x)}\right)}^{3\,ln(x)} = e^{3\,[ln(x)]^2}$$ ...
1
vote
1answer
65 views

Exponential algebra problem: Equating powers

Given Data: 5 a = 26 125 b = 676 What is the relation between a and b? I simplified 125 b = 5 4 + 5 a but how to equate a and b in above relation? Note: Answer should be in the format xa = yb ...
8
votes
4answers
162 views

Is $0^\omega=1$?

According to a definition of ordinal exponentiation defined in Kunen's Set Theory: An Introduction to Independence Proofs (pp. 26), we define $$\begin{align} \alpha^0&=1\\ ...
1
vote
2answers
62 views

Can there be a matrix $M$ such that $M^n\ne0$, but $M^{2n}=0$ for some integer $n>1$?

Reading about dual numbers, which can be modeled by a matrix $\varepsilon$ such that $\varepsilon^2=0$, I wonder if it could be generalized to something which gives nonzero square, but e.g. a zero ...
1
vote
2answers
56 views

Is this power rule true for the natural base?

Two questions 1) I was wondering if $e^{k \ln{x}}=k$ for any k. Is it? 2)To test I went to Maple and typed e^-ln(x) and it gave $e^{-ln(x)}$. I tried simplify and ...
2
votes
2answers
142 views

Convergence of $a_n= \frac{n!}{n^n}$? [duplicate]

I'm trying to find out whether sequence $a_n= \frac{n!}{n^n}$ converges or not. My textbook says to compare with $\frac{1}{n}$, and my answer sheet says that $\lim_{x\to \infty} {\frac{n!}{n^n}}\leq ...
0
votes
1answer
27 views

How do I find the value of this summation problem involving exponents?

I've looked around online quite a bit and still can't figure out exactly what to do here.
2
votes
2answers
108 views

Why does $1^{-i}$ equal 1? [duplicate]

At one point, I found an equation that works with complex logarithms, but I lost the book that contains the equation. If I feed this to Wolfram|Alpha, it states that $1^{-i}$ is equal to 1. Why is ...
1
vote
2answers
44 views

Tricky problem on function equation

How do you evaluate $$q(t)=A\cdot (2000t)^t\cdot e^{-2000t}$$ when $q(0)=4000$? I just can't get around $(2000\cdot0)^0$.
1
vote
4answers
339 views

Find the derivative of y with respect to the given independent variable

Find the derivative of y with respect to the given independent variable: $y = 3^{-x} \stackrel{D}{\longrightarrow} y' = 3^{-x} \cdot (-1) \cdot \ln 3 $ This is my teacher's solution. I don't ...
2
votes
0answers
221 views

Find the value of a^b?

Given a series as f(n)=1^1 * 2^2 * 3^3 * ......n^n since n can be very large Find the value of f(n)/f(r)*f(n-r) and output it modulo m where m is any prime. Now My approach is f(n)=1^1%m * ...
2
votes
1answer
1k views

How to deal with negative exponents in modular arithmetic?

So I think I understand how to calculate something like $(208\cdot 2^{-1})\mod 421$ using extended euclidean algorithm. But how would you calculate something like $(208\cdot2^{-21})\mod 421$? ...
0
votes
1answer
27 views

Getting rid of a fractional power over f(x)

I start with the following relation. $\frac{dy}{\sqrt{y}} = -h dt$ I then integrate it and get this function. $y^{\frac{3}{2}} = -\frac{3}{2}ht + C$ My algebra is rusty, so I'm stuck at this ...
1
vote
0answers
59 views

A variant of factorial

Given the definition of a function f as f(n)=1^1 * 2^2 * 3^3 * ... * (n-1)^(n-1) * n^n. Another function g is defined as g(n,r)=f(n)/(f(r)*f(n-r)) Given an n,r,m we are to output g(n,r)%m where m is ...
2
votes
3answers
150 views

Real analysis of powers

Show that if $a,b$ are rational numbers and $x$ is a positive real number then $x^a$$x^b$ $=$ $x^{(a+b)}$ I honestly have no idea how to even do this. Anyone have any hints or a good explanation? ...
3
votes
1answer
30 views

$y = ln(p+qe^x)/x$, solve $x$

$y = \ln(p+qe^x)/x$ $p$ and $q$ are constants. Express $x$ in terms of $y$. I believe I have to use Lambert W function, but I'm stumped. Thinking help is needed. Thank you very much!
2
votes
2answers
2k views

Why does zero raised to the power of negative one equal infinity?

I had the question of $0^{-1}$ on a math test and I naturally assumed that this evaluates to zero, but from what I have seen from various sources it is equal to infinity which I do not quite ...
2
votes
1answer
119 views

When is iterated exponentiation used and how is it defined?

I was thinking of ways to define an iterated exponentiation operation. The nice thing about addition and multiplication is that they're associative and commutative, which makes defining the sum and ...
0
votes
1answer
32 views

Periodocity of $a^{pn+q}$ mod $m$

Is $a^{pn+q}$ mod $m$ periodic? $a$, $p$ and $q$ are constants. $n$ is varied here. If it is periodic then how can I find the periodicity efficiently? Thanks in advance.