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

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1
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1answer
31 views

Find the square root of a term with a variable

I'm reviewing a PSAT score report with my son and trying to account for the College Board's answer. Below are the question and answer. I follow them as far as: $$ \sqrt{8r^2} $$ From that point, I ...
2
votes
2answers
128 views

Solutions of $a^x = x$

How can I find a bound for the solutions of the following equation without using the Lambert function? $$a^x = x,$$ where $a \in \mathbb{R}$.
2
votes
1answer
148 views

Why doesn't $2^2 = -4$?

I was just curious because a number raised to the $\frac 1x$ where $x$ is an integer greater than $1$ has $x$ solutions, why can't a number to the $x$ where $x$ is an integer greater than $1$ also ...
1
vote
1answer
111 views

Solving $a=\Big(1+\frac{b}{x}\Big)^x$ for $x$

How to solve this equation for $x$? $$a=\Bigg(1+\frac{b}{x}\Bigg)^x$$ It's not a task that I was asked to solve by someone. I just have to solve it because it's a part of my project. If it's ...
0
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0answers
83 views

Algorithm to calculate powers

Is it possible to write an algorithm that uses only multiplication and addition to calculate $a^b$ where both a and b are real numbers?
3
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1answer
106 views

$x^n y^n = (xy)^n$, proof exercise

As an exercise, I tried to prove the following theorem. Please share your thoughts about what I wrote. (The proof only uses the utensils which are listed below.) Theorem \begin{equation*} x^n ...
4
votes
1answer
170 views

Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

Conjecture: There does not exist a non-integer $x$ such that $$2^x=a$$ $$3^x=b$$ $$5^x=c$$ where $a,b,c$ are all integers. I'm aware that the similar question There does not ...
2
votes
0answers
108 views

How many zeroes would be there at the end of $11^{(5!)!}-1$?

$$11-1=10 \\ 121-1=120 \\ 1331-1=1330$$ Now it can be seen that the tens digit increases by 1 at each increment of exponent. So, only in case of $11^{10}$ the tens digit is zero and the units digit ...
0
votes
1answer
53 views

Find exponential function given two points

$f(x) = ar^x $ given that $r > 0$. I'm given two points ($3$, $\frac{8}{9}$) and ($4$, $\frac{16}{27}$). My textbook then says $$r = \frac{\frac{16}{27}}{\frac{8}{9}}$$ Why does this work? I ...
6
votes
1answer
105 views

How to prove $x^ax^b = x^{a+b}$

I am looking for a proof of one of the exponent combination laws, namely the sum of powers. Here $x, a, b \in \mathbb R$ and $x > 0$. I thought about induction but since a,b are not only positive ...
1
vote
2answers
134 views

Exponential congruence

Hi All am a bit stuck on some revision that I am trying to do. Firstly (part a) I must calculate the inverse of 11 modulo 41, which I have done and believe it to be 15. The next part is to: Now use ...
5
votes
2answers
143 views

Powers containing every digit equally often

There are several nontrivial powers containing every digit equally often, for example $32043^2$ $2158479^3$ $69636^4$ $643905^5$ $3470187^6$ A necessary condition for a power with the desired ...
3
votes
3answers
132 views

Proof that sum of first $n$ cubes is always a perfect square [duplicate]

I know that $$1^3+2^3+3^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2$$ What I would like to know is whether there is a simple proof (that obviously does not use the above info) as to why the sum of ...
0
votes
2answers
49 views

What is <any number>^i?

I think I understand what imaginary numbers are, that $i$ is basically the name we give to $\sqrt{-1}$. Does $n^i$ have any sort of meaning? Is it used for anything? You can't really multiply $n$ by ...
26
votes
2answers
340 views

Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$

After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form $\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ ...
0
votes
2answers
19 views

A perimeter and exponentiation problem

A wheel of a car lasts for 5000 scandinavian miles, and then it needs to be switched. The diameter of the car's wheel is 70 centimetres. How many laps has the wheel spinned before it needs to get ...
5
votes
2answers
88 views

Simplify $x ^ y + x ^ z$ to a formula with only one $x$

Is there a way to simplify $x ^ y + x ^ z$ to a formula with only one $x$? I know $(x ^ y)(x ^ z) = (x ^ {y + z})$, but how can it change in addition?
4
votes
7answers
398 views

Integral evaluation (step-by-step)

I'm trying to evaluate the integral by exponent. Could you help me with following steps? Integral: $$\int \frac{1}{4+\sin(x)} dx$$ $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$ $$\int \frac{1}{4+sin(x)} dx ...
15
votes
3answers
287 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
4
votes
0answers
92 views

Calculate $1^1 + 2^2 + 3^3 + … + n^n$

Is there a formula to calculate $1^1 + 2^2 + 3^3 + ... + n^n$ I searched but didn't find a formula for increasing powers
3
votes
3answers
67 views

Calculations using googolplexes

How can I calculate $\dfrac{10^{10^{100 }}}{ 10^{10^{70}}}$? I have tried using logs ie: $$\frac{10^{10^{100}}}{10^{10^{70}}}$$ $$=\frac{(100\times \ln(10)) \times \ln(10)}{(70\times \ln(10)) ...
3
votes
1answer
53 views

Find the leftmost (most significant digits) of a large exponent calculation, say $99^{99}$

I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, $99^{99} = 3.697296 \times 10^{197}$ I only need to know the digits $3697296$ Is there any ...
7
votes
3answers
155 views

How to visualize $f(x) = (-2)^x$

Background I teach Algebra and second year Algebra to middle school students. We are currently studying Exponential, Power, and Logarithmic functions. We study exponential functions (of the form ...
0
votes
2answers
32 views

Rules regarding exponents

Given the following algebra problem: $$2^{n+1}-1+2^{n+1}=2^{n+1+1}-1$$ I know $2^{n+1}=2^n2^1$ but just to confirm the truth of the problem above, I just assumed the left hand side is $2^{n+2}-1$ ...
5
votes
3answers
112 views

Rules for whether an $n$ degree polynomial is an $n$ degree power

Given an $n$ degree equation in 2 variables ($n$ is a natural number) $$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$ If all values of $a$ are given rational numbers, are there any known ...
2
votes
4answers
279 views

Calculating power without using a calculator, for example $1.05^{10}$

How to find (or estimate) $1.05^{10}$ without using a calculator? Do we have any fast algorithm for cases where base is slightly more than one? Say up to $1.1$ with tick $0.05$.
1
vote
4answers
43 views

Can we add fractional powers of negative numbers?

This question might be silly and very basic. But my friend and me happened to argue on this for long. My argument was, if $-2 \sqrt3=\sqrt{12}$ which came from $\sqrt{(-2)(-2)} \sqrt{3} $ . If this is ...
-3
votes
1answer
143 views

How to prove exponent laws for various number systems, including real exponents

How could I prove following exponent laws for set of real, in the given order? 1) $a^m*a^n=a^{m+n}$ CaseI a^m=a.a.a...to m factors a^n=a.a.a...to n factors a^ma^n=a.a.a...to m+n factors ...
0
votes
2answers
41 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 ...
2
votes
6answers
93 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 ...
1
vote
2answers
49 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 ...
33
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
448 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
54 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
31 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
105 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
26 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
40 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
12 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
46 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
41 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
3answers
43 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
47 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 ...
0
votes
2answers
30 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 ...
1
vote
1answer
92 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
27 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. ...
15
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): $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
6
votes
2answers
147 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
41 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: $1=9-8=3^2-2^3$ ...
0
votes
2answers
72 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 ...