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

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-1
votes
1answer
46 views

What is the last digit of $7^{2015}$? [on hold]

What is the last digit of $7^{2015}$?
4
votes
2answers
69 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 ...
10
votes
4answers
139 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 ...
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
71 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 ...
3
votes
1answer
44 views

What is $25^k$ + $5^k$ [on hold]

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
111 views

Exponential (to the power of a logarithm) [on hold]

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.
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
4answers
36 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?
12
votes
1answer
196 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 ...
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. ...
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 ...
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
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 ...
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
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.
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 ...
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?
-1
votes
1answer
41 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.
2
votes
2answers
29 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
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?
22
votes
10answers
636 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 ...
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.
-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 ...
0
votes
1answer
70 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 ...
6
votes
1answer
79 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, ...
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 ...
8
votes
9answers
295 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 ...
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
26 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
77 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 ...
0
votes
1answer
23 views

Exponentiating roots of unity

When given some number some complex $2n^{th}$ root of unity, $z$, how does one evaluate something such as $z^{2m}$ (with $m=kn$)? I would take $z^{2m}=(z^{2n})^k=1^k=1$, but I don't know if this is ...
2
votes
1answer
36 views

Unable to process Large numbers [closed]

A small spherical cell of diameter $1.616E^{-35}$ is exponentially multiplying as $2^n$ where n is the generation number. The duration of 1 generation is $5.39E^{-44}$ second. And the cells cluster ...
2
votes
1answer
47 views

How to calculate 10^ decimal power without a calculator?

I need to know how to calculate 10^ a decimal power, like 10^-7.4, without a calculator, in as simple a way as possible, since I will be doing questions which only allow me about a minute to a minute ...
1
vote
0answers
24 views

-1 raised to fractional indices lying between 0 and 1

For a personal project, I had to figure out what happens when $-1$ (negative one) is raised to fractional powers lying between $0$ and $1$. I thought that if I get a power $x = 0.a_1a_2a_3...a_n$ ...
8
votes
7answers
1k views

What is the right way to calculate a power?

I noticed that there are two solutions for $(-1)^{14/2}$: $((-1)^{14})^{1/2} = 1$ $(-1)^{14/2}=(-1)^7=-1$ What am I doing wrong?
5
votes
2answers
358 views

What exactly are those “two irrational numbers” x and y such that x^y is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
1
vote
0answers
49 views

Got stuck while integrating $\int x^x dx$ [duplicate]

What is the integration of $$\int x^x dx$$ And how can I understand whether an integration is possible or not? Is there any rule to understand whether a function is integrable or not?
0
votes
3answers
42 views

SAT question about integers and exponents

If $a$ and $b$ are positive integers and $$(a^\frac{1}{2}\times b^\frac{1}{3})^6=432$$ What is the value of $ab$?
-1
votes
0answers
24 views

Single element of nth power of a Matrix

I was recently solving a computational problem which had a recursive structure. With some help from the internet and a paper I found, I used mathematica to find a transformation matrix for the ...
0
votes
3answers
88 views

Finding the function that would describe this:

I'm not going to go into detail why I am interested in the next iteration of these functions, but here they are: 1: 6/(x+1) 2: 8/(2^x) 3: 10/(?) The question is, which one is next? I will say that ...
2
votes
0answers
34 views

How to define matrix power

I am currently writing a scriptum struggling with the definition of matrix power. Precisely, let $ \mathbb A \in \mathbb C^{n, n} $ and $ p \in \mathbb R $. I Currently have: If $ p \in \mathbb N $ ...
1
vote
3answers
76 views

How to derive that for every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in \mathbb{R}$ such that $y=z^x$.

I am not sure on how to derive the following statement concerning the reals (that I think should be true). For every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in ...
2
votes
3answers
89 views

Why does the minimum value of $x^x$ equal $1/e$?

The graph of $y=x^x$ looks like this: As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$. I know that $e$ is the number for ...
0
votes
0answers
21 views

Degree of a root term?

I had the textbook question: What is the degree of the following expression: $x^2\sqrt{y-5}$ Would it be 2.5, since it would be the sum of the exponents of $x$ (2) and $y$ (.5, I think)? Or ...
2
votes
1answer
62 views

Let $0<a<1$, for which $x>0$ is $x^{a^x} = a^{x^a}$ true.

Let $0<a<1$, for which $x>0$ is $x^{a^x} = a^{x^a}$ true. This is where I have gotten so far: $\log_a( x^{a^x} )= \log_a( a^{x^a} ) $ $ a^x \log_ax = x^a $ Now, I know I am only ...
4
votes
1answer
77 views

How is exponentiation defined?

Here is how I think this work: We define $ a^b = \underbrace {a\cdots a}_b $ for $a \in R $ and $b \in N$. Since so far we have not defined what $ a^{-1}$ is, $a^{-4}$ makes no sense. (right?) We ...