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

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Matrix exponent and representations of $\mathbb{R}$

It is well known that $\exp(C^{-1}AC)=C^{-1}\exp(A)C$, where $C$ is an invertible matrix. Really, $$\exp(A)=\sum{}\frac{A^k}{k!} \quad \text{and} \quad (C^{-1}AC)^k=C^{-1}A^kC,$$ so the first ...
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2answers
41 views

Simplify the expression.

Simplify the following expressions using fractional exponents, I forgot how to do this type, do I rationalize it? or can it just cancel each other out? $ {\Large \frac{\sqrt[ 5 ]{x^{ 3 }}}{ \sqrt{x} ...
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1answer
162 views

Simplify the following expressions using fractional exponents

Simplify the following expressions using fractional exponents. Display your answer using fractional exponents. $${ \sqrt[ 3 ]{x^{ 7 }} = }\text{ and }{ \sqrt[ 7 ]{x^{ 3 }} = }$$ Thanks..not sure ...
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1answer
61 views

What's the pattern? roots and powers of n?

I won't hold it against anyone if this is considered a bad question but I just don't know how else to put it and really want to know. 1st case: n = positive integer a = $\sqrt n$ b = 1 2nd case: n = ...
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0answers
92 views

Is $x^{\frac 12}$ the same as sqrt(x)

maybe this question is very simple and clear and trivial to everbody. but right now i'm not sure. the equotation $$ x^\frac{1}{2} = \sqrt{x} $$ is only true whenever $ x \geq 0 $ right? the square ...
8
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1answer
186 views

Expressing the values of a matrix at pow N

I have a square matrix (that comes from a Markov Chain) that looks like that: $$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 ...
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1answer
87 views

Systems of equations question

\begin{align*}a^a\cdot b^b\cdot c^c\cdot d^d&=\frac12\\a+b+c+d&=1\end{align*} How can we find solutions for this system of equations given that $a, b, c, d > 0$ ?
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2answers
229 views

A non-exponentially bounded analytic function?

A function $f:\mathbb R\to\mathbb R$ is said to be exponentially bounded if there is an $n$ such that for sufficiently large $x\in\mathbb R$, $\exp(\exp(\cdots \exp(x)))>f(x)$ (where the $\exp$ is ...
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1answer
1k views

What is wrong with this funny proof that 2 = 4 using infinite exponentiation?

Out of boredom, I decided to recall the following equation: $$x^{x^{x\cdots}} = 2.$$ Which, I simply rewrote like this: $x^2 = 2$, and therefore $x = \sqrt{2}$. Then I took a look at the more ...
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2answers
47 views

$P[ e^{tX} > e^{ta} ] =?$

Can anyone help me understand why given any random variable $X$, the following stands true? $$ \forall t > 0, P( e^{tX} > e^{t\epsilon} ) \le e^{-t\epsilon} E[e^{tX}]. $$ I found it in the ...
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4answers
192 views

Exponent rules with negative numbers

Hello if you could answer this please what is the difference between -2^2 and (-2)^2
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1answer
104 views

How to solve strange exponential equation?

How would equations of the form $b^x-x^a=0$ be solved for $x$, given $a$ and $b$? For instance, specifically, how would $2^x=x^2$ be solved? Does a method exist?
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1answer
40 views

How is this possible? Can someone explain?

My teacher says that $W^{2/7}B^{5/7}=1$ is equivalent to $W^2B^5=1$. Can someone explain this rule to me? Am I always able to just take the variable and raise it to the numerator of the fractional ...
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1answer
21 views

Transformation. Take n out of a root

Im kinda confused. If n is always > 0 $$ (n-a)^{\frac{x}{y}} = n*(1-\frac{a}{n})^{\frac{x}{y}} $$ is that true? Because there were some transformations in recent answers to my threads where I did ...
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1answer
34 views

Growth rate of $n^2$ vs $(\log_3(n))^3$

Which grows faster, $n^2$ or $(\log_3(n))^3$? How do I figure out which grows faster in general in these kinds of situations?
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1answer
249 views

What comes after exponents?

We use multiplication for repeated addition, and in turn use exponents for repeated multiplication. What topic comes after this, for repeated exponentials? Is there something my teachers are hiding ...
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5answers
275 views

No difference between $0/0$ and $0^0$?

I have seen discussions about both $0/0$ and $0^0$ and they differ a bit in the way that most seem ok with calling $0/0$ "undefined", while the $0^0$ discussion still seems like a dispute. If this is ...
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1answer
37 views

Question on irreducible polynomials and primes.

Consider the polynomial $p(x) = 1+\sum_{i=1}^d a_i x^i$ where $a_i$ is binary and not all $a_i$ are $0$. Is it possible that $p(2^n)$ is prime for all integer $n>-1 ?$
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4answers
70 views

Exponential algebra problem

We need to solve for x: $$54\cdot 2^{2x}=72^x\cdot\sqrt{0.5}$$ My proposed solution is below.
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3answers
216 views

How to simplify $42^{25\sqrt{25^{25}}}$?

Am a student preparing for GRE, I have no clue to solve this am attaching the screenshot of question: I need you give me a short cut or tip to deal such problems...
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5answers
958 views

Do equal bases imply equal powers?

$$x^a = x^b \Rightarrow a =b$$ So, this is a concept I used in multiple math problems and they often turn out right. The thing is, today my math teacher told me that this is not necessarily true. ...
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2answers
97 views

Power correct notation

Ok, I know this may sound dumb, but I am trying to understand which is the correct (most beauty) notation for the power function ${\rm pow}(f(x),n)$. This is the correct one: $[f(x)]^n$ From ...
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1answer
46 views

About a matrix, its powers, and a particular value…

Is it possible to find two matrices, $A$ an $B$ such that: $$A B^0 x = \begin{pmatrix} c_0 \\ ? \\ ? \\ \end{pmatrix}, $$ $$A B^1 x = \begin{pmatrix} c_1 \\ ...
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1answer
90 views

Question about powers

I've been trying to solve this problem, but I can't do it by any means other than brute force, I need help, please. The result is: 6,000.00001 $$\frac{1}{10^{-3}}+\frac{10^2}{2\cdot ...
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1answer
475 views

RSA and calculating huge exponents

I am writing an Extended Essay on RSA encryption and in the essay, I am going through a worked example of all of the stages involved (key generation, encrypting and decrypting). I am using very small ...
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1answer
92 views

Finding $x^n$ patterns

I noticed the other day while computing consecutive powers of $2$ that for $n \geq 1$, the numbers in the ones place of the values of $2^n$ repeat every 4 terms $(2, 4, 8, 6,\ldots)$. In the tens ...
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2answers
238 views

Multiplying and simplifying expressions

The expression is: $$\frac{24a^4b^2c^3}{25xy^2z^5} \cdot \frac{15x^3y^3z^3}{16a^2b^2c^2}$$ What I did was subtract the exponents of the numerator to the exponents of the denominator. I did a cross ...
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1answer
174 views

convert log(log(x)) to x-based power

I'd like to convert log(log(x)) to x-based power (I mean $x^{something}$). How can I do that?
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2answers
344 views

Under what conditions is the exponential map on a Lie algebra injective?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\exp :\mathfrak{g}\rightarrow G$ be the exponential map. In his blog, Terrence Tao notes that if a Lie group is not simply-connected, ...
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4answers
398 views

The binomial formula and the value of 0^0

Here is the text from Knuth's The Art of computer programming, 1.2.6 F formula 14: Knuth doesn't give the proof of the statement. So, I tried to write it myself. To make binomial formula equal to ...
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3answers
283 views

How to solve the integral $\int \frac{1}{e^x}\,dx$ step by step? [duplicate]

I'm primitive in integrals and derivatives and I'm trying to solve the integral $\int \frac{1}{e^x}\,dx$, but especially this integral was hard to me to solve it. So I tried: $$\begin{align} ...
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1answer
101 views

Does there exist a constant $\sqrt[4]{2} < A < \sqrt2$ such that $\lfloor A^{2^n} \rfloor$ is a practical number for all $n \in \Bbb Z^+$?

Does there exist a constant $\sqrt[4]{2} < A < \sqrt2$ such that $\lfloor A^{2^n} \rfloor$ is a practical number for all $n \in \Bbb Z^+$? I know we can exclude the range ...
12
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1answer
165 views

What is the significance of the power of $3$ in the sequence of primes given by $\lfloor A^{3^n}\rfloor ?$

Mill's constant is a number such that $\lfloor A^{3^n}\rfloor$ is prime for all $n$. The existence of such an $A$ was proven in $1947$. I know little about number theory, but I am curious as to why ...
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1answer
67 views

Find any sequence in fractional part of $e^x$?

For any infinite sequence of digits $s$, does an integer number $x$ always exist, such that the fractional part of the solution for $e^x = s$?
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3answers
116 views

Finding roots of $-3x^{1.25}-3x+10$

I'm in a math workshop, where one of the problems given was $y=–3x^{1.25} –3x+10$. Much to my frustration, the only stated way to find roots was finding x by trial and error. Is there any way to ...
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1answer
69 views

$(A^x) \mathbin\% p = (A^{x \mathbin\% (p - 1)}) % p$ if $p$ is prime. Is this true when $A$ is a matrix?

$(A^x) \mathbin\% p = (A^{x \mathbin\% (p - 1)}) % p$ if $p$ is prime. Is this property true when $A$ is a matrix? Suppose $$A=\begin{pmatrix} 1 &0 &1\\ 1 &0 &0\\ 0 &1 &0 ...
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1answer
1k views

What better way to check if a number is a perfect power?

What better way to check if a number is a perfect power? Need to write an algorithm to check if $ n = a^b $ to $ b > 1 $. There is a mathematical formula or function to calculate this? I do not ...
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2answers
208 views

What is power of number like (power of 2, power of 10)? and how to calculate power of number.

I know my question is very simple to somebody, but I'm still don't understand so far. And now my questions about this subject is: ...
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1answer
164 views

Is the operation taking a matrix to the power of another matrix well-defined?

e.g. if A and B are matrices, is there a useful definition for $A^B$? I don't see an obvious definition; but then the definition of the matrix exponential also would never occur to me independently, ...
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8answers
441 views

Do matrices have a “to the power of” operator?

Well I was sure that saying "$A^3$" (where $A$ is an $n\times n$ matrix) is nonsense. Sure one could do $(A\cdot A) A$ But that contains different operators etc. So what did my prof mean by the ...
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2answers
40 views

Decreasing power?

Say you have numbers X and Y When Y is 1 you want to times X by .5, When Y is 2 you want to times X by .5 then by .25, When Y is 3 you want to times X by .5, then by .25 then by .125, and so on. ...
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3answers
436 views

Finding unit's digit in exponentiation

Could someone please explain to me how to find the unit's digit in the following expression: $$7^{95} - 3^{58}$$
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2answers
360 views

Why is $\left(e^{2\pi i}\right)^i \neq e^{-2 \pi}$?

Here's my (obviously flawed) proof that $1=e^{-2 \pi}$: $$ 1^i=1\\ e^{2 \pi i} = 1\\ \left(e^{2\pi i}\right)^i = 1^i\\ e^{-2 \pi} = 1 $$ What's the issue? I understand that exponentiation is not ...
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0answers
59 views

Can we define root extraction using Peano Arithmetic?

I've been playing with Peano Arithmetic and I've got multiplication, division, exponentiation, and logarithms. I can't figure out root extraction but I have a stab at it. Exponentiation: $a^0 = 1, ...
2
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1answer
216 views

Number of digits and last digit of a number

How can I find the number of digits and the last digit of the number $$\large{2357^{2357^{.^{.^{.^{2357}}}}}}$$ Basically $2357$ to the power of $2357, 2357$ times.
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3answers
209 views

Does $x^{n/n} = |x|$?

Just a couple of small technical point here. If x and n are real numbers, do we have to write $x ^ {n/n} = |x|$? Or can we just reduce it to $x^{n/n} = x$? One reason I ask is because then we would ...
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1answer
641 views

Math induction problem (rules of exponents)

Hello I am doing some induction problems, I have to prove that $3^{k+1}-1$ is a multiple of 2. Suddenly they make this statement; $3^{k+1}$ is also $3 * 3^k$. Why is that?
4
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1answer
161 views

Question involving exponential tower of 19

Consider: $$ y = \underbrace{19^{19^{\cdot^{\cdot^{\cdot^{19}}}}}}_{101 \text{ times}} $$ with the tower containing a hundred $ 19$s. Take the sum of the digits of the resulting number. Again, add the ...
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4answers
131 views

Finding $\displaystyle \lim_{n \to \infty} \frac{2^{n+1} + 3^{n+1}}{2^{n}+3^{n}}$

I need help with finding $\displaystyle \lim_{n \to \infty} \frac{2^{n+1} + 3^{n+1}}{2^{n}+3^{n}}$ Thanks!
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1answer
69 views

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if a>0 and b<0

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if $a>0$ and $b<0$. How do i find the value? This doesn't make any sense.