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Questions tagged [expression-simplification]

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22 views

Multiplication/Division: How did the author simplify $(1/64)z^-3/1-4z$ to the result in the image?

I have tried multiplying the numerator and denominator by 4/z and also dividing both by 4z but alas I can't seem to get the answer. Thank you very much to anyone who answers.
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$tg^{-1}[\frac{b \sin \theta_2-a \sin \theta_1 }{a \cos \theta_1 - b \cos \theta_2}]$

Assuming $0<a<b$ and $\theta_1$ and $\theta_2$ as real numbers, is it possible to write the below expression in terms of $\theta_2$? $tg^{-1}[\frac{b \sin \theta_2-a \sin \theta_1}{a \cos \...
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1answer
20 views

How can I “fully” simplify these expanding equations? (Find a non-expanding equivalent)

I've been trying learning Algebra, but have run into the expanding equations below. After trying for a while, I've come to the conclusion that I have no idea how to "fully" simplify something that ...
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0answers
29 views

Taylor Expansion of Matrices?

I have a problem understanding a solution to a problem. Here's the background. I am looking at the criterion function for Fisher's linear discriminant function: $J(w) = \tfrac{w^tS_Bw}{w^tS_Ww}$ I ...
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1answer
87 views

Can the coefficients for the Fourier transform of the Chebyshev polynomials be further simplified?

In an article by Fokas and Smitheman, it is shown, among others things, how the finite Fourier transform of the Chebyshev polynomials can be computed. They show that, when $T_{m}(x)$ is the $m$'th ...
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1answer
20 views

Simplification of a logic equation

I am trying to simplify this logic function: $$g(x,y,z)=xy+x'z+yz$$ without Carnot (just Boole algebra). I tried going like this $$(x\cup y)\cap(x'\cup y)\cap(z\cup y)$$ and then using the ...
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42 views

Can the expression $\log2\log3\dotsm\log n$ be simplified?

$\log$ means logarithm with base $2$. When I say simplified I mean be brought into a form that is not a sum/product. Perhaps I am missing something but I did not manage to get anywhere. If it can't ...
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3answers
26 views

Solution of simple algebraic equations

Suppose $a=\alpha+2\beta$ and $b=3\alpha+5\beta$. Then by solving this, we get $\alpha=2b-5a$ and $\beta=3a-b$. I tried substitution method to solve this, but i couldn't get the above solutions viz., ...
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1answer
28 views

Simplification of a nested sum in a Taylor power series

I have this equation: $a_0+\sum_{n=1}^{\infty} a_n\cdot \left[\sum_{k=0}^{\infty}b_k\cdot x^k\right]^n$ where I know the expressions for all the $a_n$. How can I simplify the multiplication of the ...
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1answer
13 views

General formula for the following operator?

I am looking for a general formula for the following linear operator $T_n[f]$. $$T_0[f]=1$$ $$T_1[f]=\pi f(x)-\frac12$$ $$T_2[f]=2i\pi f'(x)-\pi f(x) +\frac16$$ etc. In general, $T_n[f]$ is equal ...
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0answers
31 views

Confirmation on boolean expression simplifications

This is expression no.1: $(A+B’)\cdot C + A \cdot C $. So i think these are the simplification steps: $$(A+B’)\cdot C + A \cdot C = A\cdot C + B' \cdot C + A \cdot C = B' \cdot C + A \cdot C => C \...
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40 views

Nested Tetration properties

Regarding tetration, I know properties like ${}^a({}^bn)= {}^{ab}n$ do not hold in general. When $a=b=2$, for instance, we have $$ {}^2({}^2n)={}^2\left(n^n \right)=\left(n^n \right)^{\left(n^n \right)...
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0answers
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Is there any way to get an analytic result for the distance between two circles' centers such that, in area, A∩B = (-A)∩B

In solving for the distance, d, between two circles of equal radii as a function of that radius, r, such that the area in which they overlap is 1/3 the total area, I have wound up with the following ...
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2answers
25 views

Alternative intuition behind series simplification

Given functions $$ f(x)=\sum_{i=0}^\infty a_ix^i\,\,\,\text{ and }\,\,\,g(x)=\sum_{j=0}^\infty b_jx^j $$ the following simplification $$ f(x)g(x)=\left(\sum_{i=0}^\infty a_ix^i\right)\left(\sum_{j=0}^\...
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1answer
58 views

Simplify without using complex numbers.

Simplify $$\large\sum_{r=0}^{\left \lfloor \frac{n}{3} \right \rfloor}\binom {n}{3r}$$ I tried as much as I could; tried to apply induction, tried to approach combinatorally but failed. I could not ...
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1answer
49 views

What does it mean to simplify a polynomial division?

I am doing my homework and have encountered this problem: $\frac{3x^5+5x^4+x+5}{x+2}$. I am asked to simplify it. However, when I tried to divide, it did not divide evenly, and there is a remainder. ...
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1answer
22 views

Reduction of a Power Series Solution Separated by an Integer?

I am given the following equation: $$x(x-1)\ddot y + 6x^2\dot y + 3y = 0 $$ and have solved/found the regular singular points, the exponents at singularity, the first series solution and the form of ...
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1answer
43 views

Sphere collision

I came across a problem in physics and I'm struggling a bit with getting the right intuition behind it. Below is the problem, my answer attempt and my question. Problem: Two small spheres $P$ and $Q$ ...
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3answers
49 views

How to simplify nested expression $\sqrt[3]{p + \sqrt q} + \sqrt[3]{p - \sqrt q}$

Simplify the given expression $$\sqrt[3]{a + \frac{a + 8}{3}\sqrt{\frac{a - 1}{3}}} + \sqrt[3]{a - \frac{a + 8}{3}\sqrt{\frac{a - 1}{3}}}$$ Are there any hints to do this? I have no idea
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10 views

Simplification of formula with finite set partial sums

As part of a proof I am working on, I have derived the following formula: $$f(n,P)=\frac{3^{\#P}n+\sum_{a=0}^{\#P-1}3^{a}(2^{(\sum_{b=a}^{\#P-a-1}P_{b})})}{2^{\sum P}} \\\text{where }n\in2\mathbb{Z}+1,...
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Please help with expressing variable from the inequation: $\sqrt{n+1} - \sqrt{n} < \sqrt{n+2} - \sqrt{n+1}$. [duplicate]

Can you solve it please?: $\sqrt{n+1} - \sqrt{n} < \sqrt{n+2} - \sqrt{n+1}$. I have tried to solve it but I cannot make the sqrt dissapear..
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1answer
28 views

How to simplify $\sum_{k=0}^\infty \frac{(-1)^ka^{2k+1}}{(2k-1)(2k+1)!}$?

I think that this sum can be simplified so that there's no factorial in the denominator and no exponential function in the numerator. But how can I do this? $a$ - is a constant. Could anybody show me ...
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0answers
25 views

How do I express “x” from this equation?: $y=x/(x^2+1)$ [duplicate]

How do I express "x" from this equation?: $y=x/(x^2+1)$. I am not able to express it.. What I really want to do is to find out values of this function: $f(x)=x/(x^2+1)$ if the domain is R{1,-1}. I ...
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1answer
32 views

Calculation with complex numbers

$4$ complex numbers $a,b,c$ and $d$ have the same, non-zero module $r$. Find the result of the following expression: $$|\frac{abc+abd+acd+bcd}{a+b+c+d}|$$ I think this task is supposed to be solved by ...
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3answers
84 views

Showing that $\ln \frac{1+e^x}{1+e^{-x}} = x$

Is the below a valid approach? $$\frac{1+e^x}{1+e^{-x}} = \frac{1+e^x}{1+e^{-x}} \times \frac{e^{-x}}{e^{-x}}$$ We know that $\frac{e^{-x}}{1+e^{-x}} = \frac{1}{e^x+1}$, so $$\frac{1+e^x}{1+e^{-x}} ...
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1answer
75 views

Is there a simplified way to express the following function that is defined by two Gauss' hypergeometric functions?

Consider the following function \begin{align} f_k(x) = \left( 1 + 2k \left( 1-x^2 \right) \right) {}_2 F_1 \left( \tfrac{1}{2} ,k+\tfrac{1}{2}; k+\tfrac{3}{2}; x^2 \right) - 2(k+1) \, {}_2 F_1 \...
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1answer
36 views

Simplifying Taylor series

I am solving a Quantum Mechanics problem and I came across these Taylor's series. So I was wondering, if I have simplified them correctly or is there some other option to simplify these series? I ...
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1answer
37 views

How to simplify the size of the equation using predefined variables in maple? [closed]

I am working on some simple calculations using maple for my assignment. I have an equation say $x=a$, I assigned it in maple like $$x:=a$$ Later in my calculation I got some $y$ as $$y=a^2+\sin(a)$$ ...
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4answers
56 views

Simplifying $i \left[ \ln(x+i)-\ln(x-i)-\ln(1+ix)+\ln(1-ix) \right]$

Can the following expression further be simplified and expressed in terms of usual functions such as inverse hyperbolic or inverse trigonometric functions? $$ f(x) = i \left[ \ln(x+i)-\ln(x-i)-\ln(1+...
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2answers
31 views

Logical Expression Simplification.

I'm trying to simply the expression: K = A'B'C' + AB'CD' + B'D' + C'D These are the steps I got: K = A'B'C' + ACB'D' + B'D' + C'D (Associative) K = A'B'C' + B'D' + C'D (Absorption) I know the ...
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1answer
25 views

when simplifying algebraic fractions do you go into negative powers

When simplifying algebraic fractions such as in the fraction $\frac{5p^2q}{10pq^2}$ do you go into negative powers such as stated in the rules of indices or leave it so that for this example the ...
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1answer
22 views

Proving identity related to Maxwell Stress Tensor on boundary

(electrostatic) Maxwell Stress Tensor is defined as $T_{ij}=\epsilon(E_i E_j-\frac{1}{2}|E|^2\delta_{ij})$, where $i=1,2$ and $|E|^2=E_1^2+E_2^2$ (consider two-dimensional for simplicity). The ...
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3answers
167 views

Find the exact value of trigonometric expression: $ \arctan\frac{2\cos \frac{\pi}{30} \cos\frac{\pi}{15}}{1+2\cos \frac{\pi}{30} \sin\frac{\pi}{15}}$?

How can I simplify this trigonometric expression? $$ \arctan\frac{2\cos \frac{\pi}{30} \cos\frac{\pi}{15}}{1+2\cos \frac{\pi}{30} \sin\frac{\pi}{15}}$$ I used $$\sin \frac{\pi}{15}=2 \sin \frac{...
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5answers
72 views

How do I solve $|x|\geq\frac{x}{2}+1$?

I got as far as simplifying it to $$x-2|x| \geq -2$$ What do I do next?
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1answer
48 views

Did I solve this right? Simplify $\frac{\sqrt[3]{x^2}}{\sqrt[6]{x}}$, I got $\sqrt x$

$$\frac{\sqrt[3]{x^2}}{\sqrt[6]{x}}$$ I got $\sqrt x$. The question says "Simplify the following expressions (defined in their respective domains)". The domain of the one I got isn't exactly the same ...
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1answer
36 views

Simplifying expression quartic x

Is it possible to simplify the expression: $3x^4-x^3+4x^2-5x-5$ by finding the different roots of x? I tried by the Rational Roots test, but couldn't find one easily.
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3answers
52 views

Help finding x in cubic function

In: $(x-4):(2x^3-7x^2+x+3)$ How do I find the roots of x in the cubic expression? I tried substituting the equation for 1, 3, 1:2 and 3:2, but it doesn't equal 0 in any of these cases. What should I ...
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0answers
36 views

How to Simplify this Expression for Markov Chains?

So, here's a question that has propped up in my research. Suppose that we have at our disposal $K$ independent homogeneous Markov chains evolving on a common, finite state space $S$. At each time ...
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1answer
35 views

Difference between Boolean formula and Boolean expression

I have read in a book here something I think is wrong. It says that a boolean function $\phi:B^2\rightarrow B$ defined by a truth table where {0,1} and {0,0} returns {1}, cannot be expressed with a ...
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0answers
20 views

Is it possible to simplify this expression involving modified Bessel functions of the first kind?

I need to simplify this expression: $$(w+1) I_2\left(2 \sqrt{w+1}\right)-w I_2\left(2 \sqrt{w}\right)$$ I think the context where it arose (Fourier transforms) is quite irrelevant because I am asking ...