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

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4
votes
1answer
62 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 real (or rational, for simplicity's sake) ...
6
votes
3answers
124 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 ...
2
votes
1answer
34 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 ...
31
votes
10answers
2k views
+450

Old oxford scholarship question: $a^ab^b \ge a^bb^a$

Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
-3
votes
1answer
36 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
1answer
135 views

Is it true that $n> a^2\Rightarrow n!>a^n$, $n\in\mathbb{N}, a\in\mathbb{R}$?

If so, how can it be proven? (I have evaluated it up to $n=25$.) If not, does there exist a $k\in\mathbb{R}$ such as that $n> a^k\Rightarrow n!>a^n$, with $n\in\mathbb{N},a\in\mathbb{R}$? It ...
0
votes
2answers
27 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 $2^{n+2}-1$ ...
2
votes
3answers
53 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
37 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 ...
1
vote
2answers
40 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 ...
0
votes
2answers
35 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
86 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 ...
0
votes
3answers
66 views

Simple question on exponentiation

I know this one is trivial, but I was wondering: if I have something like $$a^{b^c}$$ then i know that it should be read as $$a^{\left(b^c\right)}$$ if no other parenthesis is present. Question: if ...
4
votes
2answers
442 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
43 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 ...
2
votes
2answers
68 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
1answer
27 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
94 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? ...
1
vote
2answers
313 views

How to find out the value of numbers having fractional powers

How to find out the value of numbers having fractional powers manually without using logarithms and calculators?? For example : $2^{1.6}, 3^{2.1}, 5^{3.22}$ etc, I know we can find out the value using ...
0
votes
1answer
21 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
36 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
10 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
32 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
39 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
1answer
89 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 ...
4
votes
2answers
103 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 ...
2
votes
3answers
40 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
3answers
41 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 :)
0
votes
2answers
17 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 ...
11
votes
8answers
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): $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
4
votes
4answers
907 views

how to calculate 2^1.4

So I have got a very basic question but it didn't come up as a google search so I am posting it here. I want to know how to easy calculate 2^1.4 = 2.6390... ...
0
votes
3answers
25 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. ...
0
votes
0answers
12 views

Problem in understanding the proof of master theorem case

I am going through the proof of master method or master theroem. This is the formula that is been given by the author for the Total Work =Cn^d*(∑(a/b^d)^j) where value of J=0 to logbn as per the ...
6
votes
2answers
126 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
34 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
0answers
62 views

Sum of digits of $x^y$

Is there any simple way to calculate the sum of digits in $x^y$ other than actually computing $x^y$ and then calculating the sum? I need to calculate this for very large numbers. Please point me if ...
114
votes
16answers
9k views

Zero to the zero power - Is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...
0
votes
2answers
33 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 ...
0
votes
2answers
40 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: $$ a^p+b^q=c^r $$
2
votes
1answer
43 views

Is this version of Lagrange's four-square theorem true?

Lagrange's four-square theorem states that any natural number $n$ can be represented as the sum of four integer squares.i.e. $n = a_1\times a_1 + a_2\times a_2 + a_3\times a_3 + a_4\times a_4$ ...
0
votes
1answer
14 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
1answer
111 views

Make $a^b$ to have a complex answer [closed]

Considering I have $a ^ b$ where both are real numbers, for which values of $a$ and $b$ I will have a complex answer $(m+n*i)$. I figured out that one case is when $a<0$ and $b \in (0, 1)$. Any ...
1
vote
2answers
59 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
1
vote
3answers
458 views

Missing exponent?

$\frac{256}{2^S}=64$ How would you solve for S?
2
votes
2answers
27 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
108 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
122 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 ...
5
votes
4answers
143 views

Solving $ (x+1)^{x^2-4x+3} = 1 $ for x [closed]

Consider $$ (x+1)^{x^2-4x+3} = 1 $$ I need some hints to aid me with to problem.
2
votes
0answers
201 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
0answers
38 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 ...