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

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1
vote
0answers
28 views

If $3^{33}+3^{33}+3^{33}=3^{x}$. Solve for $x$.

If $3^{33}+3^{33}+3^{33}=3^{x}$. Solve for $x$. So we have: $$3^{33}+3^{33}+3^{33}=3^{x}$$ I added the left side and obtained: $3(3^{33})=3^{x}$ The problem I have is that extra $3$. If not, I ...
4
votes
5answers
69 views

Matrix exponential: $\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$

It is asked to calculate $e^A$, where $$A=\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$$ I begin evaluating some powers of A: $A^0= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\; ; ...
7
votes
3answers
746 views

Exponent of an exponent?

If I have an expression that gives 2^3^4, would I compute this as $(2^3)^4$ or as $2^{(3^4)}$? The two answers are wildly different. My TI gives the former but Wolfram gives the latter and I don't ...
5
votes
1answer
37 views

Let $a$ and $m$ be positive integers such that gcd$(a,m)=1$. Show that: $a^m+1$ is not a prime.

Let $a$ and $m$ be positive integers such that gcd$(a,m)=1$. Show that: $a^m+1$ is not a prime. Though I didn't check the statement with so many integers, but it looks like the equation never ...
21
votes
1answer
569 views
+50

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
0
votes
0answers
23 views

Finding n for a given P of a Bernoulli trial

I'm randomly sampling $N$ items and I want to find $n$ such that I have a probability $P$ that I'll miss one. Practically, I'd select $P$ to be something like $10^{-12}$ so I'm almost assured to ...
6
votes
2answers
67 views

Does every prime of the form $4k+1$ divide a number of the form $4^n+1$?

While playing around with Fermat's little theorem I was asking myself the question in the title and I can't answer it...
1
vote
0answers
34 views

Simplify $e^{x \cdot \log{y}}$ where $x, y \in R^N$

I'm looking to simplify the following expression (or to determine if it's even possible). Given two vectors $x, y \in R^N$, simplify $e^{x \cdot \log{y}}$. I found it in some m-code for an infinite ...
22
votes
13answers
752 views

Which is greater, $98^{99} $ or $ 99^{98}$? [duplicate]

Which is greater, $98^{99} $ or $ 99^{98}$? What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables. I don't think there is an ...
0
votes
5answers
62 views

Are these two expression equal?

My friend insisted that $(-1)^{(-n)}$ is equivalent to $(-1)^n$ for any number of $n$. A quick check in the Wolfram Alpha show ...
11
votes
4answers
9k views

How do you compute negative numbers to fractional powers?

My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure ...
15
votes
3answers
2k views

How to solve equations to the fourth power?

Is it possible to manually retrieve the value of $y$ from the following equation $$153y^2-y^4=1296$$ WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved? What I've achieved ...
3
votes
1answer
64 views

A symmetric system of nonlinear equations - how to solve?

So, I was adviced to ask a new question on my problem (as the first one wasn't very precise), that is to solve the system of equations: $$\begin{cases} x\cdot y=6 \\ x^y+y^x=17 \end{cases}$$ where: ...
0
votes
1answer
128 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 ...
5
votes
2answers
357 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
vote
1answer
77 views

If $x^{x^4}=4$. Find $x^{x^2}+x^{x^8}$

If $x^{x^4}=4$. Find $x^{x^2}+x^{x^8}$. I found this one in a competitive exam paper and found it interesting. Thanks for any help.
-1
votes
0answers
98 views

Finding the value of $(x+y)$ when $x^x+y^y=31$

As the title suggests...the question is to find the value of $(x+y)$ when $x^x+y^y=31$.Is it possible to solve this question without trial-and-error method when only this much information given.Using ...
6
votes
1answer
131 views

Does $(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$ imply $(a,b,c)=(1,2,3)$?

Question : Is the following proposition true? Proposition : For positive integers $a,b,c$ where $b\ge 2$, if $$(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$$then $(a,b,c)=(1,2,3)$. This is ...
1
vote
2answers
51 views

Simplify $\,\sqrt[10]{32a^5}$

I'm not sure if this is the correct site to ask such an elementary question but I'm trying to teach myself basic algebra and I can't understand how to do this one equation it's been so annoying. So ...
40
votes
15answers
4k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
-1
votes
4answers
45 views

Calculate the power, given all other numbers in an equation

$$100 = 200(2)^x$$ Given all numbers in the equation, how do I find $x$?
0
votes
3answers
44 views

Limit of a function raised to a power

I was working with extraction of non-electrolytic solutions and was sketching a mathematical formulae to find the limit of extracting a solvent by Nernst equation when I stumbled on this limit. ...
2
votes
4answers
113 views

Last two digits of $3^{7^{2016}}$

I need help with solving this Algebra problem: Find the last two digits of $3^{7^{2016}}$. Preferably using Euler's theorem.
1
vote
1answer
47 views

Is the following solvable for x?

I have the following equation and I was wondering if I can solve for x given that it appears both as an exponent and a base: $[\frac{1}{\sqrt {2\pi}.S}.e^{-\frac{(x-M)^2}{2S^2}}-0.5\frac{1}{\sqrt ...
2
votes
1answer
76 views

X raised to power-X raised to power-3 equals to 3.

The question is what are the possible values of $x$ when we have $$x^{x^3} = 3$$ (that is $x^3$ in the exponent itself and not $x*3$). I solved one answer by guessing that $x = \sqrt[3]3$. My work ...
0
votes
1answer
40 views

repeated exponents sign

I'm wondering if there is a exponent version of $\sum$ or $\prod$ or I've even seen a big k used for repeated division. Is there a similar symbol for exponentiation and are there any useful ...
13
votes
9answers
8k views

How to calculate $e^x$ with a standard calculator

Is there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...
1
vote
3answers
124 views

Is the solution of the equation $x^x=2$ rational?

Let $x$ be the solution of the equation $x^x=2$. Is $x$ irrational? How to prove this?
0
votes
0answers
75 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...
2
votes
0answers
67 views

Compute sum of large powers [closed]

I have the following problem. There is an array that contains values that are to be powers of $-2$. I need to calculate the sum of these powers. For example, if the array is $\{3,4,5\}$ I need to ...
3
votes
0answers
32 views

Comparing Large Exponents with different bases.

How to compare large exponents with different bases? Is there any way to roughly approximate their values? For example, sort the elements of list below based on their magnitude. ...
3
votes
3answers
113 views

Is a prime to the power of a fraction always irrational?

Let $p$ be a prime number and let $x$ be a fraction, i.e. $x \in \mathbb{Q} \setminus \mathbb{N}$. It seems to be the case that $p^x$ is always irrational. How do I prove this?
0
votes
1answer
23 views

Simple way to generate a sequence where the 1st number is $x_1$, the tenth number is $x_{10}>x_1$ and the 20th number is above some order of magnitude

I'm a bit stumped at the moment. I'm trying to generate a sequence where the first number is 1.9, the tenth is 3, and the 20th is between $10^{5-6}$ . There should be a function $f(x_1) = 1.9$, ...
-4
votes
3answers
43 views

Difference of powers of two [closed]

Is there a simple way (involving minimal calculations) to calculate $2^{987}-2^{986}=?$ Answer: $2^{986}$
2
votes
3answers
109 views

Prove that $(\sqrt3+2)^m$ is not a natural number for all natural numbers $m≥1$

The aim of this question is to show this lemma: Prove that $(√3+2)^{m}$ is not a natural number for all natural numbers $m≥1$.
0
votes
1answer
43 views

$p^3 = 2009 + 47 * 2^q$ where p and q are primes

Solve the ecuations $p^3 = 2009 + 47 * 2^q$, where $p$ and $q$ are primes. Fermat's little theorem could help.
1
vote
1answer
20 views

What kind of operation/rule was applied here?

Maybe this is a typo in our assignment and solution, but I can't tell. The question: The solution: What happened here with the minus signs in the first factor and in the exponent? Edit: the ...
1
vote
1answer
127 views

Rising/Falling Powers, Summation

1) Show that $$(-n)^{\bar p} = (-1)^n n^\underline{p}$$ (original screenshot) 2) Evaluate the sum $$\sum_{a\le n\lt b}n^{\bar p}$$ (original screenshot) Thoughts regarding question 1: I've ...
4
votes
1answer
84 views

If $x$ is a positive rational but not an integer, is $x^x$ irrational?

Let $x$ be positive, rational, but not an integer. That means $x$ can be written as $\frac{p}{q}$ with $p,q$ coprime, $p,q \neq 0$ and $q \neq 1$. Is $x^x$ always irrational? I think that this ...
2
votes
4answers
153 views

Is $x^y$ always irrational if $x$ is rational and $y$ is irrational?

Prove or disprove: "If $x$ is a rational number, and $y$ is an irrational number then $x^y$ is irrational" I am stuck with this, these are my steps. let $x=2$ and $y=\sqrt{2}$ ...
0
votes
1answer
21 views

Will the absolute logarithm always produce the correct real result if one exists?

I'm a computer scientist, so my math skills are a bit rudimentary. The application I'm writing is more or less about solving equations. I'm only interested in real number solutions, so imaginary ...
2
votes
4answers
335 views

Show that $(\sqrt{2}-1)^n$ is irrational

Show that for all $n\in \mathbb{N}$ the number $(\sqrt{2}-1)^n$ is irrational. I do not get the idea of the proof at all, any help appreaciated. edit: I am also thinking whether it will be ...
0
votes
1answer
80 views

Can $\sqrt{a}^\sqrt{b}$ be rational if $\sqrt{a}$ and $\sqrt{b}$ are irrational?

Let $a$ and $b$ be rational numbers, such that $\sqrt{a}$ and $\sqrt{b}$ are irrational. Can $\sqrt{a}^\sqrt{b}$ be rational? I found examples, where the irrational power of an irrational ...
1
vote
2answers
46 views

Simplification of $n^{1/\sqrt{\log n}}$

I would like to simplify this function, how can we do it ?
4
votes
7answers
135 views

How do i convince students in high school for which this equation: $2^x=4x$ have only one solution in integers that is $x=4$?

I would like to convince my student in high school level using a simple mathematical way to solve this equation: $$2^x=4x$$ in $\mathbb{z}$ which have only one integer solution that is $x=4$ . ...
7
votes
4answers
187 views

Why can the integral $\int_{x=0}^{\infty} x\mathrm{e}^{-\alpha x^2}\mathrm dx$ not be evaluated by parts?

Can the integral $\int_{x=0}^{\infty} x\mathrm{e}^{-\alpha x^2}\mathrm dx$ be evaluated by parts to show that $\int_{x=0}^{\infty} x\mathrm{e}^{-\alpha x^2}\mathrm dx= \frac{1}{2\alpha}$ I know that ...
1
vote
2answers
62 views

series function

We know that there are some series that can be written in short, for example: $$ \sum_{n=0}^\infty x^n=\frac{1}{1-x},\qquad |x|<1 $$ Is there similar function for $$ \sum_{n=1}^N x^{1/n} $$ or $$ ...
0
votes
4answers
115 views

Why is $(-1)^x=e^{i\pi x}$

I was recently taught exponentials and I decided to play around with negative bases, which they told me were not allowed. The obvious place to start was negative one, and, as expected, the graphing ...
25
votes
10answers
3k views

What is the accepted syntax for a negative number with an exponent?

A friend is taking a college algebra class and they are teaching him that $$-3^2 = -9$$ Their explanation is: $$-3^2 = -(3^2) = -9.$$ It has been a long time for me but I thought that in the ...
2
votes
2answers
483 views

How to find out the value of numbers having fractional powers

How to find out the value of numbers having fractional powers manually without using logarithms and calculators?? For example : $2^{1.6}, 3^{2.1}, 5^{3.22}$ etc, I know we can find out the value using ...