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

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2
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4answers
98 views

Basic Mathematics. Trouble with powers and polynomials.

Greets, I'm hoping to complete the exercises in chapter 1.3 Basic Mathematics, Serge Lang. The section question is: Expand the following expressions as sums of powers of $\;x\;$ multiplied by ...
5
votes
6answers
139 views

Basic Mathematics. Trouble with powers.

Greets, In Chapter 1.3, Basic Mathematics, Serge Lang, there is the question: Express each of the following expressions in the form $2^m3^na^rb^s$, where $m, n, r, s$ are positive integers. b) ...
3
votes
1answer
462 views

taking the log of $a^b$ (Project Euler problem 29)

I've been stuck on Project Euler problem 29 and thus asked a friend who solved it how to do it. What he basically did was for each power was: $\left(\frac{\log_{10}(a)}{\log_{10}(2)}\right)\cdot b$ ...
3
votes
2answers
663 views

Do inequalities hold under square-root (or exponentiation in general)?

This has been bothering me lately. My proof-skills are rusty (and were never great to begin with). I dimly recall having seen this (or something related to it) in a math course I took a while ago, but ...
3
votes
0answers
134 views

Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia $$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k ...
1
vote
1answer
78 views

Solving an equation with $x$ as powers

How would I go about solving $$2^x -2^{x-2}=3 *2^{13}$$Hints please. Thank you.
3
votes
6answers
156 views

Motivation for creation of complex exponentiation

I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived?
2
votes
3answers
77 views

why if x in 1/n power >(<) y in 1/m power then x in c/n power >(<) y in c/m power?

As you might guess this is one more stupid question from non-matematician, and you are right. I found this exercise in "Algebra and trigonometry book": $7^{1/2}$ or $4^{1/4}$. After some googling I ...
8
votes
3answers
2k views

Why is 1 raised to infinity Not defined and not “1” [duplicate]

$1$ square is $1$, so is raised $1$ to $123434234$. My maths teacher claims that $1$ raised to infinity is not $1$, but not defined. Is there any reason for this? I know that any number raised to ...
0
votes
1answer
373 views

A calculator's solution to irrational exponent

An irrational number cannot be represented by $\frac{p}{q}$ where $p$ and $q$ are integers. And when we encounter exponents with decimal points, it is a possible way and a rather simple one to turn ...
6
votes
6answers
491 views

How is this proof flawed?

$\sqrt{x}=-1$ $\sqrt{x}^2=(-1)^2$ $x=1$ Now substitute it into the original equation $\sqrt{1}=-1$ $1=-1$
1
vote
1answer
57 views

Can we write $\sqrt[w]{z}=z^\frac{1}{w}$ when both $w$ and $z$ are complex numbers? [duplicate]

Let $w$ and $z$ be complex numbers defined in terms of real numbers $a$, $b$, $c$ and $d$ as follows: $$ w = a+bi \\ z = c+di $$ Can we analogically write $$ \sqrt[w]{z} = z^\frac{1}{w} \qquad ...
12
votes
1answer
342 views

Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324…$ is irrational?

(somewhat related to my earlier question) Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$). The ...
1
vote
6answers
741 views

What is the value of $2^{3000}$ [closed]

What is the value of $2^{3000}$? How to calculate it using a programming language like C#?
30
votes
1answer
604 views

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
4
votes
1answer
65 views

Simplifying $y=2^{2/3} + 2^{-1/3}$

I am working on a calculus problem where I have to find the local minimum. The value I got was $$y=2^{2/3} + 2^{-1/3}.$$ I simplified it and got this: $$ y=2^{2/3} + \frac{1}{2^{1/3}}$$ ...
2
votes
1answer
130 views

Definite integral including the ratio and power functions of a single variable

I find trouble in calculating the following integral: $$ \int_0^R \frac{m\cdot x}{m+s\cdot x^a} \,dx $$ Mathematica does not provide an output for this function, however, there seems to be an output ...
1
vote
4answers
144 views

Complex power of a complex number: Find $x$ and $y$ in $x + yi = (a + bi)^{c+di}$

$$ x + yi = (a + bi)^{c+di} $$ Find $x$ and $y$ in terms of $a$, $b$, $c$ and $d$. Where, $i$ is defined as $\sqrt{-1}$ and $a$, $b$, $c$, $d$ are real numbers. I defined two new real number ...
2
votes
1answer
57 views

Formula/Algorithn for Exponential factoring?

Given $s = a^b$ find $a$ and $b$. my first algorithm was the obvious brute force method of checking all $b$ roots or dividing by all possible $a$. But I am wondering if there is a more efficient ...
0
votes
1answer
44 views

Finding that probability of the event is small

Let $x_1, \ldots, x_n$ be Bernoulli random variables with the probability of success $P(x_i=1)=p$. Let $\epsilon>0$. Show that probability $$ P\left(\left|\sum_{i=1}^nx_i-p\right|> ...
4
votes
4answers
986 views

Can you raise a Matrix to a non integer number? [duplicate]

So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
1
vote
2answers
1k views

Trace of the matrix power

Say I have matrix $A = \begin{bmatrix} a & 0 & -c\\ 0 & b & 0\\ -c & 0 & a \end{bmatrix}$. What is matrix trace tr(A^200) Thanks much!
3
votes
2answers
124 views

Division with negative exponents

I have a problem that looks like this: $$\frac{20x^5y^3}{5x^2y^{-4}}$$ Now they said that the "rule" is that when dividing exponents, you bring them on top as a negative like this: ...
10
votes
5answers
625 views

Evaluating tetration to infinite heights (e.g., $2^{2^{2^{2^{.^{.^.}}}}}$)

The Problem How can you evaluate (i.e., get a value for) Tetration (i.e., iterated exponentiation) to infinite heights? For example, what would be the value of this expression? $$ ...
0
votes
1answer
186 views

De Moivre's theorem question.

State De Moivre's theorem and use it to find integers $ A,B,C$ such that $$\sin^5\theta=A\sin5 \theta + B\sin3\theta + C\sin\theta.$$ I know De Moivre's theorem, how to prove it, and converting to ...
0
votes
1answer
62 views

Put the following in rectangular form.

$$(\sqrt{3}+i)^7$$ My question: $r = 2$. For $\theta$, do I use $\dfrac{\pi}{6}$ or $\dfrac{\pi}{6} + 2n\pi$? The book uses the former but I thought the latter is more appropriate. Thank you.
0
votes
1answer
76 views

Does $ \ (g^a Mod\ p)^b\, $ $\equiv$ $ \ (g^a)^b (Mod\ p)\, $ hold true?

Are these two equations: $$ \ (g^a Mod\ p)^b\, $$ $$ \ (g^a)^b (Mod\ p)\, $$ one and the same? If yes then how And if no then how to solve the first equation?
4
votes
2answers
246 views

How to evaluate powers of powers (i.e. $2^3^4$) in absence of parentheses?

If you look at $2^{3^4}$, what is the expected result? Should it be read as $2^{(3^4)}$ or $(2^3)^4$? Normally I would use parentheses to make the meaning clear, but if none are shown, what would you ...
-1
votes
3answers
151 views

Find the smallest natural number that satisfy $13^N = 1 \pmod {2013}$

Moderator Note: This is a current contest question on Brilliant.org. Find the smallest natural number that satisfy: $$13^N = 1 \pmod {2013}$$ My idea is to use the Fermat's little theorem ...
-1
votes
1answer
74 views

$n^0 = 1$ ? Try for this case? [duplicate]

We know that anything to the power $0 = 1$ i.e. $n^0 = 1$ My question is that, is $0^0 = 0$ or $1$ and why?
0
votes
0answers
36 views

check validity of following manipulation

in my algebra book,there is written following well known identity $e^{2*\pi*i}=1$ generally we can use also this identity $e^{k*\pi*i}=(-1)^k$ and if instead of $k$,we put $2$ we get ...
9
votes
7answers
4k views

Pattern to last three digits of power of $3$?

I'm wondering if there is a pattern to the last three digits of a a power of $3$? I need to find out the last three digits of $3^{27}$, without a calculator. I've tried to find a pattern but can not ...
2
votes
4answers
112 views

If $3^x \bmod 7 = 5$, what is $x$ and how?

I am an amateur java programmer who is stuck on this problem: $$3^x \bmod 7 = 5$$ then what is $x$ and how? If you can even explain the method for how to arrive at the solution, then it will be very ...
1
vote
0answers
73 views

Are there other powers which follow the rule $a^b = b^a$ than $2^4$? [duplicate]

I was trying to find these powers, but to my disappointment I only found $2^4 = 4^2$. Edit: $a$ must be different to $b$ of course. Is that the only possible setting, and why? If we assume the number ...
2
votes
3answers
124 views

How to find this expression $1000! \mod 3^{300}$

How to find this expression $(1000!\mod 3^{300})$?
0
votes
1answer
40 views

Exponent, logarithmic question

I'm reading an article related to bioinformatics and I found this formula: Probability of $x =(1-y/n)^t$ or approximately $e^{-yt/n}$. My question is how do we pass to the approximation given in the ...
0
votes
4answers
123 views

Square and square root and negative numbers [duplicate]

Are they equal? -5 = $\sqrt{(-5)^2}$
3
votes
4answers
83 views

Proof of $\sqrt{2^{2^k}} = 2^{2^{k-1}}$?

It's quite easy to observe that for $k \ge 0$: $$ \begin{align} 2^{2^k} &= 4, 16, 256, 65536, \dots\\ \sqrt{2^{2^k}} &= 2, 4, 16, 256,\dots \end{align} $$ More in general: $$ \sqrt{2^{2^k}} ...
3
votes
1answer
137 views

Exponential Sum - Solve for X

I'm wondering if there is a way to solve for x given the following equation: $$A^x + B^x = C^x$$ where A, B, and C are known constants. For pythagorean triples, $x = 2$. I've seen a lot of stuff for ...
3
votes
3answers
62 views

Triplets 4th of power of first equals to sum of other two

I was looking through the admission test for the University Normale of Pisa and I found a problem that I don't know how to solve, it state something like that: Find all the triplets of number (x, y, ...
10
votes
3answers
851 views

What's the difference between $3^{3^{3^3}}$ and $27^{27}\;$?

Why does $\;\large3^{3^{3^3}}\;$ evaluate to a larger number than $\;\large 27^{27}$?
0
votes
1answer
148 views

For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?

I came across the following question: For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$? What I have tried: Since for $ \alpha = 1 $ is clearly non-differentiable in ...
0
votes
1answer
27 views

exponentiation question

I have a homework question on exponents, the question asks "simplify the expression and eliminate any negative exponents" The Question is as follows $$(2x^2y^4)^3(3x^{-3}y)^2$$ and my working out ...
1
vote
5answers
396 views

Generate solutions of Quadratic Diophantine Equation

Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: How to solve Quadratic Diophantine Equation Here's the answer: $$ ...
0
votes
2answers
78 views

$ 10^{-9}[2\times10^6 + 3^{1000}] $

$$ 10^{-9}[2\times10^6 + 3^{1000}] $$ I'm stuck on solving this. I wasn't able to put this into my calculator since the number is too big for it to calculate. So far I've done this: $$ ...
2
votes
2answers
3k views

Raising a square matrix to a negative half power

I want to implement the following formula (taken from Kaiser, 1970) in R where $R$ is square matrix of correlations: $$S = (\textrm{diag } R^{-1})^{-1/2}$$ I understand the diagonal and inverse ...
9
votes
2answers
295 views

Algebraic equation problem - finding $x$

$$(x^2 +100)^2 =(x^3 -100)^3$$ How to solve it?
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3answers
136 views

conditions under which real-matrix exponential are equivalent

Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$? Thanks!
1
vote
2answers
50 views

Calculating z^n without trigonometric functions

I'm looking for a formula to calculate z^n for complex z and integer n. I know that I can use the Moivre's formula but I will use the formula in a computer program ...
16
votes
6answers
3k views

A question comparing $\pi^e$ to $e^\pi$ [duplicate]

I was doing an algebra problem set following a chapter on logarithms and exponentiation, and it presented this "bonus question": Without using your calculator, determine which is larger: $e^\pi$ ...