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

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0
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0answers
25 views

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

All the following terms are positive integers. If $$(i_1)^{a_1}(i_1+1)^{b_1}=n $$ $$(i_2)^{a_2}(i_2+1)^{b_2}=n $$ $$(i_3)^{a_3}(i_3+1)^{b_3}=n $$ then $n$ is a perfect power, i.e. it is expressible ...
0
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3answers
18 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
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0answers
10 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 ...
4
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8answers
948 views

How to determine without calculator the biggest of two numbers

How can you determine which one of these numbers is bigger (without calculating): $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
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2answers
97 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
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2answers
27 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
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2answers
23 views

Using the basic laws of exponent [on hold]

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 ...
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2answers
37 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: $$ a^p+b^q=c^r $$
4
votes
1answer
61 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 ...
-1
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1answer
32 views

Basic definition of exponent and $a^0$? [closed]

How can we define $a^0$ as being equal to $1$? Why?
2
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1answer
39 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
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1answer
13 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
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 ...
2
votes
1answer
110 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 ...
5
votes
2answers
105 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
120 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 ...
6
votes
4answers
137 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
3answers
136 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
43 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: $${\left(e^{ln(x)}\right)}^{3\,ln(x)} = e^{3\,[ln(x)]^2}$$ ...
0
votes
0answers
57 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 ...
1
vote
1answer
21 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
120 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
49 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
3answers
43 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 ...
0
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1answer
14 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
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2answers
93 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
votes
0answers
24 views

calculate high power of 10

I need some help with this calculation : $$10^{73099226}\mod n $$ I am calculating it in python but that says **Result too large **
1
vote
2answers
36 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
25 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
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0answers
199 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
29 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 $(208\cdot2^{-21})\mod 421$? ...
-1
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0answers
12 views

Solving For A Given Equation With Exponents

I'm unable to solve the following equation. The question asks: The population $p$ at time $t$ years is assumed to be: $p= {2800ae^{0.2t} \over 1+ae^{0.2t}}$ where $a$ is a constant. Given that $p ...
0
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1answer
23 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
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0answers
43 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 ...
0
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3answers
35 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
21 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
38 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
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0answers
37 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 ...
2
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5answers
45 views

How to solve large modulus manually? [closed]

given $52^{35}\mod 85$ I know I can change that to $52\cdot 52^{2^{16}}\mod 85$ but I'm unsure where to go from there.
0
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0answers
19 views

Limit of power function

I want to be sure if that is correct, Could you please point me to some good references for that kind of function : When $a \to +\infty$ $ x^a =\begin{cases} 1 & x = 1\\ \\ +\infty & x>1 ...
0
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0answers
25 views

How are the power of 2 and square roots consistent?

This is something that keeps bothering me. We all know that $x^{2} = |x|^2$ And we know that, e.g., $\sqrt{9} = 3$ But we can also use $\sqrt{x} = x^{\frac{1}{2}}$ and $(x^{a})^{b} = x^{ab}$ ...
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1answer
29 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.
1
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0answers
37 views

Commuting exponentials of non-commuting matrices

For two non-commuting matrices $A,B \in M(2,\mathbb{K})$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, can be shown that: $$ e^C=e^{A+B}=e^Ae^B=e^Be^A \iff \begin{cases} ...
0
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1answer
18 views

Converting to Power of Ten Representaion

For very large calculations Wolfram Alpha offers a variety of different representations of the number. One of these is the number written in the form $10^{10^n}$, where $n$ is usually some long ...
1
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1answer
68 views

Quaternion exponential

Given an imaginary quaternion $ \mathbf{v}=\alpha \mathbf{i}+ \beta \mathbf{j}+\gamma \mathbf{k} $ its exponential is: $ e^\mathbf{v}=\cos \theta +\mathbf{v}\dfrac {\sin \theta}{\theta} $ where ...
2
votes
1answer
47 views

What's bigger, the sum of powers or the power of the sum?

Do we know if $(\sum\limits_{i=1}^n a_i)^k \geq \sum\limits_{i=1}^n a_i^k$ for any $k\geq1$?
2
votes
3answers
80 views

Solving $\ln(x) = e^{-x}$

I'm trying to solve $\ln(x) = e^{-x}$ but I can't really get how to do it :( (Removing a statement that was incorrect, as explained by the comments below) Additionally, while I started to solve it I ...
0
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1answer
50 views

rational exponent of negative base

I have the definite integral $$\int_{1}^{\,9} {\frac{6}{\sqrt[3]{x-9}}}\, \mathrm dx$$ When I try to evaluate it I get the indefinite integral equals $9(x-9)^{2/3}$ and evaluating at the limits gives ...
1
vote
2answers
45 views

Rules of i ($\sqrt -1$) to a power

$i^{2014}$ power =? A. $i^{13}$ B. $ i ^{203}$ C. $i^{726}$ D. $i^{1993}$ E. $i^{2100}$ I don't understand the concept that powers of i repeat in fours and that "two powers of i are equal if ...
0
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0answers
46 views

A basic doubt to compute exponential of a matrix

Given a matrix I want to evaluate $e^{A}$. The method suggested uses the taylor expansion. But, it is also written that the method works well if the largest and smallest eigen values are not well ...