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

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2
votes
2answers
44 views

Does the complex modulus satisfy the power identity $|z^r|= |z|^r$?

Can we "split the modulus" of complex numbers? Let $z\in\mathbb{C}$. Then, does $$|z^{r}|=|z|^r$$ hold, where $r\in\mathbb{R}$. Is this true even for $r\in\mathbb C$ ? Also, can we show this? I am ...
2
votes
2answers
124 views

Could you solve $x↑^n2=x↑^m2$?

As my title asks, could you solve $x^2=2^x$? But that's the worrisome part, as I noticed $x↑^n 2=x↑^m 2$ and $2↑^p x=2↑^q x$ will always have a solution at $x=2$. However, there is bound to be at ...
3
votes
1answer
20 views

Question concerning comparison of different tetration functions

Let $a_{1}=2$, $a_{n+1}=2^{a_{n}}$ for $n \geq 1$ Let $b_{1}=3$, $b_{n+1}=3^{b_{n}}$ for $n \geq 1$ Is is true that $a_{n+2}>b_{n}$ for all $n \geq 1$? If so, is the proof elementary? (Use only ...
2
votes
1answer
36 views

Proving $a^b$ is well defined

How do I prove that $$\lim_{(m,n) \to \infty} a_m^{b_n} = a^b$$ where $a,b \in \mathbb R$, $a_i,b_i \in \mathbb Q$, $a_m \to a$, $b_n \to b$ and $a$ and $b$ are not both zero, and $a_m >0$ I can ...
0
votes
0answers
26 views

Do quaternions linearise tetration [on hold]

This is just a wild guess as I am not very familiar with quaternions or tetration. Sorry if my terminology is not quite right, I hope you get the general idea. Complex exponentiation is linear in ...
-2
votes
1answer
39 views

Proof: Raising a complex number to a rational power

The problem from the textbook is: Prove that if (a complex number) $z$ is a number on the unit circle, then $z$ has finitely many distinct powers $z^n$ if and only if the argument of $z$ is a ...
14
votes
1answer
237 views

Proving that $e^\pi=e^{-\pi}$

I've been stuck with this for a while now. I have this chain of reasoning that would imply $e^{-\pi}=e^\pi$, obviously false, since $e^\pi$ and $e^{-\pi}$ are two real distinct numbers and so I must ...
1
vote
3answers
179 views

Will $a^a$ ever out-grow $9^{9^{^\ldots}}$?

I am trying to come up with the largest finite number that can be made using a set number of characters. I have two expressions which are calculated and printed out by a program (theoretically - they ...
10
votes
4answers
201 views

Why isn't $e^{2\pi xi}=1$ true for all $x$?

We know that $$e^{\pi i}+1=0$$and $$e^{\pi i}=-1$$ So$$(e^{\pi i})^2=(-1)^2$$$$e^{2\pi i}=1$$ Because $1$ is the multiplicative identity,$$(e^{2\pi i})^x=1^x$$$$e^{2\pi xi} =1$$should also hold ...
4
votes
2answers
82 views

Last digit of $235!^{69}$

Problem What is the last digit of $235!^{69}$? It's been far too long since I did any modulo calcuations, and even then, the factorial would set me back. My initial thought goes to the last digit ...
-1
votes
1answer
51 views
8
votes
5answers
1k views

Taking the square root of an imaginary number

We know that when we take the square root of a negative real number, it's realness "splits open" and an "imaginary" dimension is introduced (characterized by the presence of iota). The question is, ...
2
votes
1answer
2k views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
0
votes
1answer
41 views

Does $2^{k+1} = 2^k * 2^1$?

I'm not sure how to deal with an exponent like this. Can I simplify it into terms that are easier to work with? I know that $2^3 · 2^4 = 2^{3+4} = 128$, but I don't know about $2^{k + 1}$
7
votes
3answers
85 views

Is there an irrational number $a$ such that $a^a$ is rational?

It can be proved that there are two irrational numbers $a$ and $b$ such that $a^b$ is rational (see Can an irrational number raised to an irrational power be rational?) and that for each irrational ...
8
votes
9answers
302 views

Why $e^x$ is always greater than $x^e$?

I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$ I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the ...
24
votes
10answers
662 views

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
6
votes
2answers
175 views

Last digits of power towers $7$, $7^7$, $7^{7^7}$, $7^{7^{7^7}}$, … don't change, and generalisation

While playing around with Wolfram Alpha, I noticed that the last four digits of $7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did ...
3
votes
1answer
44 views

What is $25^k$ + $5^k$ [closed]

This is an extremely simple problem, but I can't find an example anywhere for some reason. I know that $30^k$ is not correct But I have no idea what else makes sense.
0
votes
3answers
114 views

Exponential (to the power of a logarithm) [closed]

How do I solve the following equation: $(3x)^{ln3}=(4x)^{ln4}$ Thanks in advance!
0
votes
2answers
25 views

Question on double inequality with radicals

A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$? Thank you very much.
28
votes
6answers
2k views

What is the order when doing $x^{y^z}$ and why?

Does $x^{y^z}$ equal $x^{(y^z)}$? If so, why? Why not simply apply the order of the operation from left to right? Meaning $x^{y^z}$ equals $(x^y)^z$? I always get confused with this and I don't ...
0
votes
4answers
37 views

Simple Fraction needing explanation

$$\frac{x}{x^{-1/2}} = x^{3/2}$$ How? I don't see what is going on here. What rule is being used to achieve this amount?
1
vote
2answers
65 views

How to calculate $\lim \limits_{h \to 0}{\frac{a^h-1}{h}}$?

As the title says, I would like to prove for $f(x) = a^x$ there is always some constant c such that $f'=cf$. Is calculating the limit the right approach to solve this problem? Also, how to show there ...
0
votes
2answers
37 views

On analogy between $\Bbb Z$ and $\Bbb F_q[x]$

There are objects and operations analogous between $\Bbb Z$ and $\Bbb F_q[x]$. For example primes in $\Bbb Z$ and irreducibles in $\Bbb F_q[x]$ are analogous and so is multiplication operation. ...
12
votes
10answers
1k views

Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$

I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no ...
48
votes
9answers
4k views

Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
1
vote
1answer
28 views

Integer Y with N Repeating Digits of X?

I have a single Base 10 digit X. I want to return number Y where Y is digit ...
1
vote
2answers
72 views

Evaluating limit $\lim_{n\to\infty}({\sqrt{4^n + 3^n} - 2^n})$

I have to find: $$\lim_{n\to\infty}\left({\sqrt{4^n + 3^n} - 2^n}\right)$$ I plugged in some numbers and it seems as if this sequence were approaching infinity, but I do not know how to begin ...
2
votes
3answers
689 views

What is the rule of equating exponents called?

For example: $$2^{2n-1} = 2^{n+2} \Rightarrow 2n - 1 - n - 2 = 0 \Rightarrow n = 3$$ I couldn't find this rule in properties of exponents i.e when the bases are equal, the exponents can be equated. ...
1
vote
1answer
16 views

Prove the following simple exponentiation equality.

Having trouble with the following proof. Given $b > 1, c > 0$, prove that $ \exists \; x$ s.t. $b^{x} < c$. We can't use $log$, and I have already shown that $b^{x} > c$ by using the ...
2
votes
4answers
78 views

Finding $\lim_{n \to \infty} \dfrac{n^n}{(2n)!}$

Struggling to apply Squeeze THM to find this limit. Specifically, I need a sequence which is always larger than the one in the problem, but which can easily be derived from the middle sequence.
2
votes
2answers
31 views

Common factor out from a sum of exponential functions

From the below equation, which is a sum of two exponential functions I would like to compute the common factor $n$ $$ d = \exp\left(\frac{-x}{n}\right)+\exp\left(\frac{-y}{n}\right)$$ Unfortunately, ...
0
votes
1answer
33 views

Modular exponentiation and two primes

Given two primes $11$ and $5$, find all $\alpha> 1$ such that $$\alpha^{5} \equiv 1 \pmod{11}$$ What theorem will help me to find it out?
0
votes
1answer
24 views

How can I find the exponent of a power of two from its remainder modulo a power of three?

Suppose that ${2^m}\equiv k\pmod {3^n}$ and that I know $n$ and $k$. Is there a way to find the lowest (or indeed any) value for $m$ other than by enumerating the possibilities? Note: I'm aware that ...
-1
votes
1answer
44 views

Why $5x \cdot 8x - 3x \cdot (-3x) \ne 49^4$?

$$5x \cdot 8x - 3x \cdot (-3x) = 40 x^2 + 9x ^2 = 49 x ^2 ,$$ but why is it not $49^4$? I just need this clarified and the rule behind this.
0
votes
1answer
26 views

Equation solution in modular arithmetic

Given two primes $11$ and $5$, find all $α > 1$ such that $α^{5} \equiv 1 \mod 11$. How would you compute that?
0
votes
1answer
72 views

Infinite tetration of $i$

Proof Euler's identity; $$e^{i\pi} + 1 = 0$$ can be manipulated in order to obtain the result: $$e^{i\pi} = -1$$ Raising both sides of the equality to the power of $i$ gives, after ...
1
vote
0answers
50 views

Why can't the exponent laws be extended to complex numbers?

$(x^a)^b=(x^b)^a=x^{ab}$ only when real numbers are involved. This implies that the base must be a positive number. For example, with $x=-1$, $a=2$, and $b=\frac{1}{2}$, ...
7
votes
5answers
110 views

How can I compare the numbers $2^{39}$, $5^{19}$ and $52^7$?

I have to compare the numbers $2^{39}$, $5^{19}$ and $52^7$. I don't know how to do that because their exponents don't have anything in common.
6
votes
3answers
433 views

Algorithm for tetration to work with floating point numbers

So far, I've figured out an algorithm for tetration that works. However, although the variable a can be floating or integer, unfortunately, the variable ...
-1
votes
1answer
30 views

How to represent a $5$ digit number that has $62$ choices per digit?

If you have a $5$ digit number that can be 0-9A-Za-z how would you represent that? total_number_of_records = 5 digits * (10 + 26 + 26) ^ 5 I want to find out ...
4
votes
3answers
3k views

upper bound of exponential function

I am looking for a tight upper bound of exponential function (or sum of exponential functions): $e^x<f(x)$ when $x<0$ or $\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$ Thanks a ...
6
votes
1answer
80 views

Prob. 6, Chapter 1 in Baby Rudin [duplicate]

Here's problem $6$ in Chapter $1$ in the book Principles of Mathematical Analysis by Walter Rudin, $3$rd edition: Fix a real number $b$, such that $b > 1$. $(a)$ If $m, n, p, q$ are integers, ...
0
votes
1answer
632 views

How to find the common base of terms in an expression?

I'm teaching myself basic algebra from a book and am stuck on a question. In the current section it is about expressing numbers as powers of the same base. So $9$ maybe expressed as $3^2$. Another ...
1
vote
1answer
40 views

if $|f(x)-f(y)|\le |x-y|^{\sqrt 2}$ then is $f$ a constant function?

if $f: \mathbb{R}\to \mathbb{R}$ satisfies $$|f(x)-f(y)|\le |x-y|^{\sqrt{2}}$$ for all $x,y\in \mathbb{R}$ ,then is f increasing ,decreasing or constant? in my view ,it is clear that $|f(x)-f(y)|$ is ...
2
votes
2answers
53 views

How can I tell which one of these numbers is greater?

I have two very large numbers how do I tell which one is greater. The two numbers are $$\sum^{9(10^{99}) } _{i=1}i^9$$ and $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$
0
votes
3answers
27 views

How to deal with the different powers of the same variable here?

In this solution they take out the ro to put it on one side of the equation and give it power 6 How do we take out ro as a common when the two ro's have different powers? Thank you.
1
vote
1answer
27 views

Solving a modular exponentiation problem

How do I solve for $y$ in this congruence: $$11^{112111} \equiv y \bmod 113$$ I saw that $113$ is prime and so by Fermat's Little Theorem, it means $a^{112} \equiv 1 \bmod 113$. $$11^{112111} ...
0
votes
2answers
81 views

Simplify $(2^{2015})(5^{2019})$

Question : $(2^{2015})$$(5^{2019})$ How do I simplify/solve that without a calculator? I have no idea how to continue, I know it's important to get the Base number the same so I can add the ...