Questions on fractions, which are expressions (not values) of the form $\frac pq$.

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6
votes
5answers
139 views

What is $\frac{x^{10} + x^8 + x^2 + 1}{x^{10} + x^6 + x^4 + 1}$ given $x^2 + x - 1 = 0$?

Given that $x^2 + x - 1 = 0$, what is $$V \equiv \frac{x^{10} + x^8 + x^2 + 1}{x^{10} + x^6 + x^4 + 1} = \; ?$$ I have reduced $V$ to $\dfrac{x^8 + 1}{(x^4 + 1) (x^4 - x^2 + 1)}$, if you would ...
1
vote
1answer
9 views

How to simplify equations with a constant times a polynomial to a power

I've started doing integration in class and I often come across parts like this towards the end of my integration: $\frac{2(6x^2+6x+9)^{3/2}}{3} +C$ And I get stuck. How should I go about ...
0
votes
1answer
40 views

Simplifying an expression.

I've been trying to simplify this expression for quite a long time but I just can't get to the result given in the book. The result in the book says that this expression should be simplified to (a*b; ...
-4
votes
2answers
86 views

Find all values of $x$ such that $\frac{x-4}{2x-3}$ is an integer. [closed]

Find all values of $x$ such that $\frac{x-4}{2x-3}$ is an integer. I tried many ways to do it, like by setting it to a fraction and what not, but it could just come for me, I hope you can help. ...
0
votes
0answers
25 views

Inverse Laplace Transform by Partial Fraction Expansion

I've been trying to solve this partial fraction for a Laplace transformation but I can't. Is there any way to solve it? $$\frac{(s-t)^2}{((s-t)^2-1)((s+1)^2+4)}$$ Could somebody help, I've been ...
2
votes
2answers
35 views

What is a field with bounded whole and unbounded fractional part called?

Given numbers of the form: $W + \frac n d$ where $d \gt n \ge 0, W \ge 0$, and all are integers, when defining addition and multiplication on these numbers, I want $W$ to be bound by a positive ...
0
votes
2answers
51 views

Simplification of $\sum_{k=1}^n \frac{1}{4k^2-1}$

So I want to simplify this expression: $$\sum_{k=1}^n \frac{1}{4k^2-1}$$ and Wolfram Alpha tells me it can be simplified to two forms: $$\frac{n}{2n+1}$$ and $$\frac{1}{2}-\frac{1}{2(2n+1)}$$ The ...
0
votes
0answers
15 views

Sum of finite number of terms of the series $\sum\limits_{L=2}^{L_{max}}\frac{1}{e^{i \phi / L} -1}$

Good day everyone. Are there any chances to get a compact formula for the following sum of finite number of terms? $$\sum\limits_{L=2}^{N} \dfrac{1}{e^{ \frac{i \varphi}{L}} -1}$$ N and $\varphi$ ...
0
votes
1answer
22 views

Factors that Impact a Weighted Average the Most

I am trying to determine how to calculate the factors that have the most significant impact on a weighted average. For example, let's say I am reviewing the number of patients that responded to a ...
3
votes
4answers
99 views

Why Does $f(x) = x\sqrt{x+3}$ Only Have One Critical Point?

I am trying to find the critical points of the function $f(x) = x\sqrt{x+3}$, then by using the First Derivative Test, determine which ones are a local maximum, local minimum, or neither. Using the ...
3
votes
2answers
124 views

Difference between fractions at group level have different sign than difference between fractions in aggregate

I have obtained a result (perhaps incorrectly; we shall find out) that appears paradoxical. Suppose I am interested in comparing fractions between 'groups' (not in the strict mathematical sense) ...
0
votes
1answer
14 views

Conditions present after solving for x

The question asked me to solve $\dfrac {x^2+2x-8}{x^2-x-2}=3$ The answer is $x=0.5$, which I worked out, but in the answers it says that I must state the condition that $x≠2$ to get full marks. Why ...
0
votes
1answer
32 views

Problem in solving an Integral.

I'm solving a solid of revolution problem and I'm stuck at this point. The u-subtitution doesn´t work, I don´t know what method use. $\pi\int(\frac{4x-1}{8x^4})^2$
1
vote
2answers
43 views

Tips and tricks for speedy mental division with harder fractions

I'm looking for a strategy to solve these kind of questions rapidly. Do you guys have any suggestions? $$\frac{\frac{15}{25}}{X}=\frac{14}{35}.$$ A. $\frac{3}{5}$ B. $\frac{63}{30}$ C. ...
0
votes
1answer
18 views

Find a common denominator

I'm trying to integrate a function but first I need to find a common denominator for: $\frac{A}{x-8}\ $ + $\frac{B}{x+1}\ $ + $\frac{C}{x-1}\ $
1
vote
2answers
52 views

Finding average of denominator knowing average of numerator and average of fraction [closed]

Good day to you all. I have a little problem I have been banging my head on for a while. I have come to think that it is impossible, but I hope you can save me. I have a fraction, $ ...
3
votes
3answers
103 views

Proving that the sum of fractions has an odd numerator and even denominator.

I'm struggling to show that, for all $n>1$ $$ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{k}{m} $$ where $k$ is an odd number and $m$ is an even number. Proof: The proof is by ...
2
votes
1answer
29 views

why this Diophantine equation $2ks=(5t+3)(16t+9)$ has always a solution for every $k$?

I would like to solve the following Diophantine equation and show that it has always a solution; i.e. for every positive integer $k$, there exists an integer $t$ such that the fraction is an integer: ...
0
votes
1answer
15 views

Convert mibibyte-days to gibibyte-months.

I have a web host that charges $1 for every Gibibyte (1024 MiB) that I use per month. Currently I am using about 7.6 MiB (Mibibytes) per day. I tried to do this myself and ended up with hopelessly ...
5
votes
4answers
540 views

Summing reciprocal logs of different bases

I recently took a math test that had the following problem: $$ \frac{1}{\log_{2}50!} + \frac{1}{\log_{3}50!} + \frac{1}{\log_{4}50!} + \dots + \frac{1}{\log_{50}50!} $$ The sum is equal to 1. I ...
0
votes
1answer
39 views

How to cancel out a negative in a denominator?

The question was to make $y$ the subject in $x=5-3y$ (i.e. solve for $y$). My working was this: $$\begin{align*} x&=5-3y \\ -3y&=x-5 \\ y&=(x-5)/(-3) \end{align*}$$ But this was ...
4
votes
2answers
26 views

Distributing multiplication of rational functions

I am having trouble distributing with fractions. This $$ \left(\frac{1}{(x + 3)} + \frac{(x + 3)}{(x - 3)}\right)\, (9 - x^2) = -\frac{(x^2-3)}{9-x^2}(9-x^2) $$ has the answer $\left\{\left[x = - ...
0
votes
2answers
33 views

Rules for solving inequalities with negative fractions

What are the algebraic rules for solving inequalities with negative signs and fractions like: $$\frac{1}{x}<-\frac{1}{5}$$
3
votes
4answers
223 views

Write a formula as a sum of fractions with constant numerators

I'm supposed to write this formula: $$\frac {9a + 43}{a^2 + 9a + 20}$$ As a sum of fractions with constant numerators as: $$\frac {7}{a+5} + \frac {2}{a+4}$$ The first step is of course: $$\frac ...
0
votes
1answer
22 views

Rearranging exponential functions

Using two different strategies, I've derived an equation for a particular function $f(\phi)$. That equation is $$ f(\phi)=\frac{1}{1-e^{-T\gamma}}(1-e^{-T\gamma\phi}). $$ However, the paper whose ...
0
votes
2answers
27 views

fraction multiplication - having trouble with picture

I have this problem where 3/4 x 2/3 and answer is 1/2 however, I am having hard time finding the correspondent picture from the attach file. Is the answer H or O? if yes why? I am helping my son and ...
2
votes
1answer
33 views

What is the best way of resolving an expression with square roots in denominator?

I was resolving the following question from my textbook: Write each of the following expressions as a single fraction, simplifying your answer where possible: $4 - \frac{1}{\sqrt{12}} + ...
0
votes
1answer
47 views

How to Find the Domain of The Inverse of (4(e^x)-5)/(25(e^x)+12)

These are the steps I have taken so far: In order to find the inverse of the function, I did the following steps: ...
-2
votes
3answers
48 views

Adding 1 to a/b?

I've been learning algebra 2 and came across a question telling me to add 1 to $\frac{a}{b}$ by finding the least common multiple of the divisors. So I got $\frac{1b}{1b}+\frac{a}{b}$ that simplified ...
0
votes
1answer
11 views

fraction simplication, resolution with one variable

I'm having some problems to understand a simplification and I know the correct answer but I'd like to understand why my method is wrong or what rule I'm breaking. I have an equation like the ...
3
votes
4answers
95 views

Express the following product as a single fraction: $(1+\frac{1}{3})(1+\frac{1}{9})(1+\frac{1}{81})\cdots$

I'm having difficulty with this problem: What i did was: I rewrote the $1$ as $\frac{3}{3}$ here is what i rewrote the whole product as: ...
0
votes
1answer
89 views

Subtracting or Multiplying Fractions 3/4-1/2

This is the given scenario to help visualize the problem at hand. A bird feeder is filled with 3/4 of a full bag of seeds. The birds ate 1/2 of what was in the bird feeder. What fraction of a full ...
0
votes
2answers
20 views

Inequality relationship when additing a constant to the denominators

When $\frac{a}{b}>\frac{c}{d}$ where $a, b, c,$ and $d$ are positive real numbers, is $\frac{a}{b+1}>\frac{c}{d+1}$ true?
1
vote
1answer
71 views

Reasoning and Explanation for $2/3-1/2$

Denise says that $2/3-1/2 = 1/3$ and gives the reasoning indicated in Figure (below) to support her answer. Is Denise right? If not, what is wrong with her reasoning and how could you help ...
0
votes
1answer
26 views

Adding fractions with variables and using common denominator? Rather basic math, but here I am..

So I have these fractions I need to merge together and shorten: $$\frac{1}{2a+8} + \frac{4}{a^2-16} + \frac{4}{a-4}$$ I understand that I need to make the denominators the same so that I can merge ...
2
votes
1answer
40 views

Simplify the fraction with radicals

I want to simplify this fraction $$ \frac{\sqrt{6} + \sqrt{10} + \sqrt{15} + 2}{\sqrt{6} - \sqrt{10} + \sqrt{15} - 2} $$ I've tried to group up the denominator members like $ (\sqrt{6} + \sqrt{15}) ...
-1
votes
1answer
18 views

Simplify complex fraction

This is a really low level question. I'm trying to simplify $$f(x) = \frac{36x^{-2} -3x^{-1} -18}{12x^{-2} -25x^{-1} +12}$$ After factoring, removing negative exponents, and flipping the second ...
0
votes
1answer
8 views

Removing additive factors from the denominator of a fraction

Suppose I have a variable $x_L$ defined as follows: $$ x_L=\frac{r_1e^{-t_0s}}{s + r_2 + r_3}, $$ Is there a way for me to rearrange this fraction to get something in this form $$ x_L = ...
1
vote
1answer
50 views

Rationalise the denominator and simplify $\frac {3\sqrt 2-4}{3\sqrt2+4}$

Does someone have an idea how to work $\dfrac {3 \sqrt 2 - 4} {3 \sqrt 2 + 4}$ by rationalising the denominator method and simplifying?
2
votes
1answer
35 views

Inequality for an alternating sum of binomial coefficients times a fraction

I need to show that $$\sum_{n=0}^x(-1)^n {x\choose n}\left(\frac{1}{n+l-1}-\frac{1}{n+l}\right) >0,$$ for any $l>0$. I tried to prove this, but I didn't get anywhere. Apparently the above ...
0
votes
1answer
24 views

Simplify the fraction

I want to simplify this fraction $ \frac {1+a+a^2+...+a^n}{1+a^{-1}+a^{-2}+...+a^{-n}} $ where, $ a \in {\mathbb R^*} $ and $ n \in {\mathbb N^*} $ I think is some kind of formula here
2
votes
2answers
99 views

What is the name for the mathematical property involving addition or subtraction of fractions over a common denominator?

For any number $x$ where $x\in\Bbb R$ and where $x\ne0$, what is the mathematical property which states that: $${1-x^2\over x} = {1\over x} - {x^2\over x}$$
0
votes
1answer
35 views

Simplifying this expression, trigonometry

I have been having trouble understanding how $$6-6\cos\left(\frac{\pi}{4}\right) = 3\sqrt{2}.$$ My main problem is the conversion of the two separate terms into one.
0
votes
2answers
25 views

Derivatives of a fraction function

An example of a fraction function is: $$y= \frac{-8x}{(x^2 + 3)^2}$$ The quotient rule says that if the function one wishes to differentiate, $f(x)$, can be written as: $$h(x) = \frac{f(x)}{g(x)}$$ ...
2
votes
1answer
22 views

Summation and Patterns Question

I have a really urgent question. Today i was looking at this Pattern (Using fractions) : $\frac{1}{1*3}+\frac{1}{3*5}+\frac{1}{5*7}+\frac{1}{7*9}+...$=? The pattern is $\frac{n}{2n+1}$. I was ...
0
votes
1answer
54 views

Simplifying a fraction with radicals

Simplify the following fraction as much as possible $$\def\A{\sqrt{15}}\def\B{\sqrt{35}} \frac{(8+2\A)^{3/2}+(8-2\A)^{3/2}}{(12+2\B)^{3/2}-(12-2\B)^{3/2}} $$ This problem is just driving me ...
1
vote
2answers
48 views

How to compare fractions without finding common denominators?

This is the question: Use reasoning other than finding common denominators, cross-multiplying, or converting to decimals to compare each pair of fractions listed below. Which is greater? Give ...
1
vote
1answer
55 views

What is the value of $x$ which satisfies: $\frac{17}{85}+\frac{19}{95}+\frac{21}{105}+\frac{23}{115}+\frac{25}{125}+\frac{x}{135}=1$

What is the value of $x$ which satisfies: $\frac{17}{85}+\frac{19}{95}+\frac{21}{105}+\frac{23}{115}+\frac{25}{125}+\frac{x}{135}=1$ I thought $x=27$ because the numerator seems to have a pattern of ...
0
votes
6answers
81 views

How to prove $\frac{1}{n} > \frac{1}{n+1}$?

I understand this fact is very simple, but how would I go about it from a fundamental perspective only know that $n$ is less than $n+1$?
3
votes
2answers
164 views

Inequality - GM, AM, HM and SM means

I've got stuck at this problem : Prove that for any $a > 0$ and any $b > 0$ the following inequality is true: $$ {3} {\left(\frac{a^3}{b^3} + \frac{b^3}{a^3}\right)} \geq \frac{a}{b} ...