Rational Definition and 628 Threads

Homework Statement Is the group of positive rational numbers under multiplication a cyclic group. Homework EquationsThe Attempt at a Solution So a group is cyclic if and only if there exists a element in G that generates all of the elements in G. So the set of positive rational numbers would...

$\tiny{Q5|8.5.17}$ $\textsf{Evaluate}$ \begin{align*}\displaystyle I&=\int_0^{9}\frac{x^3 \, dx}{x^2+18x+81} \color{red}{=243\ln2-162} \end{align*} OK even before I try to get this answer trying to see what road to take u substituion long division and remainder partial fractions (the...

$\tiny{242t.08.02.09}$ $\textsf{ Evaluate to nearest thousandth}\\$ \begin{align*} \displaystyle I_{8.1.31} &=\int_5^{10} \frac{3x^5}{x^3-5} \, dx =885.576\\ \end{align*} $\textit{ok tried getting to this answer by: }$ $u=x^3-5 \therefore du=3x^2 \, dx$ $\textit{but it didnt seem to go to good}$

Hi I've been tying to figure this out for days. The answer has to be in the form of a rational equation. John wants to start a padlock company. The production cost of one padlock is 12$. John plans to spend 15 000$ on advertising. He decides to sell each padlock for 20$. 1.Which equation gives...

Homework Statement I'm trying to understand negative bases raised to rational powers, when calculating principle roots for real numbers. I'm not worried about complex solutions numbers at this stage. I just can't find a concise explanation I can understand anywhere. I'm self learning as an...

$\tiny{206.07.05.88}$ \begin{align*} \displaystyle I_{88}&=\int\frac{1}{(x+2)\sqrt{x^2+4x+3}} \, dx \\ &=\int\frac{1}{(x+2)\sqrt{x^2+4x+3+1-1}} \, dx \\ &=\int\frac{1}{(x+2)\sqrt{(x+2)^2-1}} \, dx \\ u&=(x+2) \therefore du=dx \\ I_{88}&=\int\frac{1}{u\sqrt{u^2+1}} du \end{align*} $\textit{so...

Homework Statement Sketch the graphs of the following functions and show all asymptotes with a dotted line y = (2x - 6)/ (x2-5x+4) i) Equation of any vertical asymptote(s) ii) State any restrictions or non-permissible value(s) iii) Determine coordinates of any intercept(s) iv) Describe the...

Calculation of $\displaystyle \int e^x \cdot \frac{x^3-x+2}{(x^2+1)^2}dx$

$\textsf{evaluate}$ \begin{align} \displaystyle {I}&={\int{\frac{x+2}{x^2+1}dx}}\\ &=\int{\frac{x}{x^{2}{+1}}dx{\ +\ 2}\int{\frac{1}{x^{2}{+1}}}}{\ }dx\\ u&=x^{2}+1 \therefore \frac{1}{2x}du=dx\\ x&=\sqrt{u-1}\\ \end{align} ...? $\textit{calculator answer.?}$ $\dfrac{\ln\left(x^2+1\right)}{2}...

I am trying to find primitives to the rational function below but my answer differs from the answer in the book only slightly and now, I am asking for your help to find the error in my solution. This solution is long since I try to include all the steps in the process. The problem $$ \int...

Hello! This is my first post on these forums. So I was stuck with this question in my Mathematical Analysis exam, and it is as follows: ƒ(x) = 0 if x ∉ ℚ and (p + π) / (q + π) - (p / q) if x = (p / q) ∈ ℚ (reduced form). 1- Prove ƒ is discontinuous at all rational numbers except 1: This is...

Problem 1 Simplify/solve: 2*81/2-7*181/2+5*721/2-50 Attempt at solution: a1/2=√a ⇒ 2*√8 - 7*√18 + 5*√72 - 50 = 2√8 - 7√18 + 5√72 - 50 = ? Do not know how to proceed beyond this point. Have experimented with little luck. Problem 2 Simplify/solve: a-1(1+1/a2)-1/2 * (1+a2)1/2 Attempt at...

Homework Statement Must find equation meeting these requirements: 1. Hole at x=5 2. Vertical Asymptote at x= 0 3. Horiztonal Asymptote at g(x) = 2 4. Y-int at (-1.5, 0) Homework EquationsThe Attempt at a Solution I know that there must be an (x-5) in both the numerator and denominator due to...

Homework Statement Person 1 can paint 3m2 in x hours, person #2 can paint 7m2 in x + 2 hours however together they can paint 22m2 of the wall in 10 hours. How many hours did person #2 take to paint the wall? Homework EquationsThe Attempt at a Solution Im confused at how to start the question...

Homework Statement teacher gave me the correct solution which said that the result from him was as follows ##\frac{1}{a^2b^6}## Homework Equations original problem was as follows ##\frac{ (\frac{1}{a})^2*b^{-3}}{ ab^3 }## The Attempt at a Solution...

Homework Statement A polynomial, P(x), is fourth degree and has all odd-integer coefficients. What is the maximum possible number of rational solutions to P(x)=0? Homework Equations P(x) = k(x-r1)(x-r2)(x-r3)(x-r4) P(x) = 0 when x = {r1, r2, r3, r4} The Attempt at a Solution I expanded the...

We have the rational function : $$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\left(\frac{1-ix}{1+ix}\right)^{n/2}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$ It's not hard to prove that ...

$\Large{S6.7.R.19}$ $$\displaystyle I=\int\frac{x+1}{9{x}^{2}+6x+5}\, dx =\frac{1}{18}\ln\left({9{x}^{2}+6x+5}\right) +\frac{1}{9}\arctan\left[{\frac{1}{2}\left(3x+1\right)}\right]+C $$ $\text{from the given I thought completing the square would be the way to solve this} \\$ $\text{but I...

$\tiny\text{LCC 206 {r3} integrarl rational expression}$ $$\displaystyle I=\int \frac{3x}{\sqrt{x+4}}\,dx =2\left(x-8\right)\sqrt{x+4}+C \\ \text{u substitution} \\ u=x+4 \ \ \ du=dx \ \ \ x=u-4 \\ \text{then} \\ I=3\int \frac{u-4}{\sqrt{u}} \ du = 3\int {u}^{1/2}du -12\int...

$$\displaystyle \int\frac{{x}^{2}}{\left(4-{x}^{2}\right)^{3/2}} \ dx$$ $u=4-{x}^{2} \ \ \ du=-2x dx \ \ \ x=\sqrt{4-u}$ $$\displaystyle \int\frac{4-u}{ \left(u\right)^{3/2}} -2 \sqrt{4-u}\ du$$ Stuck

nmh{1000} $\tiny{\text {Whitman 8.7.28 integral rational expression}} $ $$\displaystyle \int\frac{t+1}{{t}^{2}+t-1}\ dt$$ $\text{book answer}$ $$\displaystyle\frac{5+\sqrt{5}}{10} \ln\left({2t+1-\sqrt{5}}\right) +\frac{5-\sqrt{5}}{10} \ln\left({2t+1+\sqrt{5}}\right)+C$$ $\text{expansion}$...

Hi, I downloaded the file: 7.0.0.4-RATL-RRENT-WIN-all-FP04. Its size is: dir 7* Volume in drive D has no label. Volume Serial Number is B269-AF38 Directory of D:\download 06/04/2016 10:40 PM <DIR> 7.0.0.4-RATL-RRENT-WIN-all-FP04 06/04/2016 09:14 PM 131,195,766...

if for rational a,b,c $ax^3+bx+c=0$ one root is product of 2 roots then that root is rational

Hey guys, I am not to good in Math and I am having issues solving this equation. 2x-9/3x + 8 = 3/7x How do you solve this equation? The answer is supposed to be x = 0.3956?

9/se^{2}-4 = 4-5s/s-2

In The Martian we see airlocks that an SUV would fit into. Wastes air every time it's opened, even if pumped down. I suggest we make an airlock that clamshells down on a space suit and leaves the minimum amount of room between suit and lock for air to be reclaimed from. You could have banks of...

Homework Statement Show that the set ##\{x \in \mathbf Q; x^2< 2 \}## has no least upper bound in ##\mathbf Q##; using that if ##r## were one then ##r^2=2##. Do this assuming that the real field haven't been constructed. Homework Equations N/A The Attempt at a Solution Attempt at proof: ##r\in...

I ran across this years ago. It’s defined by the addition rule: a/b + c/d = (a+c)/(b+d) (Sounds crazy – until you think of x/y as meaning There’s a pile of x things on one side, and one of y things on the other.) Applying it in almost a Pascal's Triangle sort of way generates ... er, a...

the question is: Rewrite the rational equation y=(-5x-18)/(x+4) to show how it is a transformation of y=1/x. describe transformations looks like it is shifted 4 to left, then stretched by factor of -5x-18. Is that accurate? would you elaborate beyond that?

Homework Statement Solve ##y=\mathrm{exp}(\frac{-x\pi}{\sqrt{1-x^2}})## for x when y = 0.1 Homework Equations ##\mathrm{ln}(e^x)=x## The Attempt at a Solution ##\mathrm{ln}(0.1)=\frac{-x\pi}{\sqrt{1-x^2}}## ##(\frac{-\mathrm{ln}(0.1)}{\pi})^2=\frac{x^2}{1-x^2}##

Well, let's look at how this works. Quadratic equations can have either 1, 2, or no zeroes. If it has no real zeroes, the zeroes it DOES have are complex, so that's obviously not it. Let's imagine ax^2 + bx + c = 0 has one zero, call it \alpha (Cuz it looks pretty). Then that means ax^2 +...

I want to convert a recursive real formula to rational number representation, but I get the wrong response. For the real formula: k = 1.9903694533443939 u1 = -12.485780609032208 u2 = -6.273096981091879 u3 = k * u2 -u1 /// 1st iteration u3 = -1.7763568394002505E-15 // approx. zero /// 2nd...

Homework Statement Find the Laurent expansions of ##f(z) = \frac{z+2}{z^2-z-2}## in ##1 < |z|<2## and then in ##2 < |z|< \infty## in powers of ##z## and ##1/z##. Homework Equations Theorem: Let ##f## be a rational function all of whose poles ##z_1,\dots , z_N## in the plane have order one and...

Homework Statement Prove that for all n ≥ 1, there exists a polynomial f(x) ∈ ℚ[x] with deg(f(x)) = n such that f(x) is irreducible in ℚ[x]. Homework Equations In mathematics, a rational number is any number that can be expressed as the quotient or fractionp/q of two integers, a numeratorp...

m and n are integers. log2(i) = m/n 2^(m/n) = i 2^m = i^n 2^0 = i^4 = 1 so that means that log2(i) is rational because there are integers n and m so that log2(i) = m/n , they are m=0 and n=4. But what I do get about this proof is that it seems to imply that log2(i) = 0/4 = 0 while google says...

in every radixial representation, except of course for those cases in which the numerator is a factor of some natural-number power of the radix? For the radixial system we know (i.e., because we are bilateral and have arms that have 5 fingers), this would mean that any possible natural number...

Homework Statement Here the formulation of Dedekind's axioms that I am using:Suppose that line ℓ is partitioned by the two nonempty sets ##M_0## and ##M_1## (i.e., ##\ell = M_0 \cup M_1##) such that every point between two points of ##M_i## is is also in ##M_i##, for ##i = 0,1##. Then there...

Homework Statement I would like to compute the following integral: I = \int\limits_0^\pi \mathrm{d}\theta \, \frac{\sin^2 \theta}{a^2 + b^2 - 2 \sqrt{ab} \cos \theta} where ##a,b \in \mathbb{R}_+##. 2. The attempt at a solution Substitution ##x = \cos \theta## yields I = \int\limits_{-1}^1...

There is a question that has been puzzling me for quite a few years by now, since the moment I've heard that quarks have charge ##-e/3##, ##2e/3## etc. There seem to be 2 independent answers to the title: 1. There is a Charge Censorship Principle, where you can never observe a charge ##e/3## and...

I am not suicidal. But I am in my 70s and I am trying to make plans and policies appropriate for my age, dealing with end of life. I'm doing this while I'm still healthy. I wear a Medic Alert ID Tag that says "Refuse all Medical Care" PF is visited by civil intelligent people so I would...

How can the properties of rational exponents be applied to simplify expressions with radicals and rational exponents?

Euler mentions in his preface of the book "Foundations of Differential Calculus" (Translated version of Blanton): I don't understand here, who/who all had invented/discovered the study-of-ultimate ratio (differential calculus) for rational functions long before (Newton and Leibniz), without...

1/2x-7/x-3 Subtract

Homework Statement (-64)^(3/2) Homework Equations None. The Attempt at a Solution There is no answer that can be reached and it is supposed not be a real number. I was wondering why that is. How is it that there is no "real" answer to this problem?

$$\lim_{{x}\to{\infty}}\frac{\sqrt{3x^2 - 1 }}{x-1}=\sqrt{3}$$ I tried dividing everything by ${x}^{2}$ but not

To find the horizontal asymptotes of a rational function, we find the limit as x goes to infinity. Given the rational function ##\displaystyle\frac{x + 1}{\sqrt{x^2+1}}##, we can find the limit by multiplying the numerator and the denominator by ##\frac{1}{x}##. This gives us ##\frac{1 +...

Given that we have the expression ##\displaystyle-\frac{1}{(x-2)(x-2)(x-3)}~\cdot~\sqrt{\frac{(x-2)^{2}}{(x-3)(x-1)}} ##, how do we simplify it, step by step? Specifically, I am concerned about the ##\sqrt{(x-2)^{2}}## term. Are we allowed to cancel this with the ##(x-2)## in the denominator?

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with an aspect of the proof of Lemma 3.70 ... The relevant text from Rotman's book is as follows:In...

Do you use two lines (of regular ruled paper) for a rational function? FOr example, would you use two lines or one for the equation ## f(x) = (x^2-3) / x+2 ## I can see three ways (in general) of writing this. 1) numerator and denominator each get one line 2) num and den are written smaller...

The rational numbers $a,\,b,\,c$ (for which $a+bc$, $b+ac$ and $a+b$ are all non-zero) satisfy the equality $\dfrac{1}{a+bc} =\dfrac{1}{a+b}-\dfrac{1}{b+ac}$. Prove that $\sqrt{(c−3)(c+1)}$ is rational.