Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, radicals, etc.

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3answers
57 views

I need some basic introduction to limits

So, I know you can obviously cut out a value if it is multiplying and dividing something at the same time, right? Like: $$\frac{4h-2xh-h^2}{h} = \frac{h(4-2x-h)}{h} = 4-2x-h$$ But then I saw this ...
0
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1answer
74 views

Basic question about simplifying a square root

I just wanted to know how to get from $\sqrt{12}$ to $2\sqrt{3}$ Because my buddy was teaching me math the other day and gave me a list with some basic exercises to do, one of which is to solve ...
3
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2answers
46 views

Question about arc length.

I am having some trouble finishing an arc length problem. Specifically, what is $\int_{0}^{1}|x'(t)| dt=?$ Is it just $\int_{0}^{1} |x(t)| dt=|x(1)-x(0)|$? If so why?
0
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0answers
46 views

Proof that $ k^2<2^k$ [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $k\geq 5$, prove that $k^2<2^k$. I assumed that $k^2<2^k$ I want to show that $(k+1)^2<2^{k+1}$ The ...
1
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1answer
93 views

Solving $[x]+[x]=[2x]$

Solving the equation $[x]+[x]=[2x]$ Since $[x]$ is the greatest integer function. I tried, $\forall x\in\mathbb{N}$, we have $[x]=x$ and $[2x]=2x$ this implies that $[x]+[x]=[2x]$, but if ...
1
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2answers
579 views

Sum of the reciprocals of divisors of a perfect number is $2$?

How do I show that the sum of the reciprocals of divisors of a perfect number is $2$? I tried $d_i\mid n$ with $i\in\mathbb{N},\;d_i\leq n$ then ...
2
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6answers
333 views

Proof that $n^3-n$ is a multiple of $3$. [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $n$, prove that $n^3-n$ is a multiple of $3$. I assumed that $k^3-k=3r$ I want to ...
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3answers
85 views

How does these square roots work? [closed]

$$\tag{1}2\sqrt{90} - 5\sqrt{160} + 3\sqrt{250} - 2\sqrt{40} = ?$$ $$\tag{2}\sqrt[3]{a\cdot b^2} \cdot \sqrt[4]{a^3 \cdot b \cdot c^2} = ?$$ $$\tag{3}\frac 6{2 \cdot \sqrt{3} - 3} = ?$$ picture
17
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4answers
29k views

Factorial, but with addition [duplicate]

Is there a notation for addition form of factorial? $$5! = 5\times4\times3\times2\times1$$ That's pretty obvious. But I'm wondering what I'd need to use to describe $$5+4+3+2+1$$ like the ...
0
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1answer
39 views

If $U = (1, 2)$, $V = (3, -4)$, is the answer to $2U + \frac{1}{2}V$ the vector $(3.5, 2)$? Check my answer.

If $U = (1, 2)$, $V = (3, -4)$, is the answer to $2U + \frac{1}{2}V$ the vector $(3.5, 2)$? I did the following: \begin{align*} U : ( ( 2 * 1 ), ( 2 * 2 ) ) &= ( 2 , 4 )\\ V : ( ( 0.5 ...
5
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4answers
12k views

What's the algebraic property where you can flip the fractions in an equation?

Earlier in algebra, we spent over 20 minutes trying to figure out $$ \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e} \,\,\,\, \text{solve for }R_2 $$ when the teacher said "What you start out with is ...
6
votes
2answers
305 views

struggle simplifying $\sqrt{9+\sqrt{5}}$

I need to simplify $\sqrt{9+\sqrt{5}}$ I already do this (proven it) $\sqrt{9-4\sqrt{5}}=2- \sqrt{5}$ But I couldn't when apply to ...
0
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1answer
55 views

How to calculate sum?

As you can see in my screenshot, I take 25% from score and sum the result which come up to 64. My question is that how can I obtain the same 64 from sum of full(175) and score (144)? I have tried ...
0
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1answer
2k views

Time and Work in Unitary Method

Let us suppose, A and B can do a given work in 12 and 18 days respectively. They work alternately for equal period of time. And A started the work. Now, what is the time taken by A and B to complete ...
5
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1answer
81 views

$x;y;z\in \mathbb{Z}$ such that $(x-y)(y-z)(z-x)=x+y+z$. Prove that : $27\mid x+y+z$

Let $x;y;z\in \mathbb{Z}$ such that $(x-y)(y-z)(z-x)=x+y+z$. Prove that : $27\mid x+y+z$ Thanks :) P/s : I don't have any ideas about this problem..!!
0
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3answers
46 views

Adding and subtracting roots

I have to find (a/b), b=(1-(√3/3)), a= (1+ (1/√3)) I know I can just punch this into my calculator. However, my teacher says we need to know how to add these and subtract these but just as said that, ...
2
votes
2answers
64 views

show that $ \forall n \in \mathbb N$, $9\mid\left(10^n + 3\cdot4^{n+2} +5\right)$

Using congruence theory, show that $ \forall n \in \mathbb N$, $9\mid\left(10^n + 3 \cdot 4^{n+2} +5\right)$. The proof is quite simple with induction, but how can it be proved with congruences?
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3answers
207 views

how to solve factorial involving multiplication

I am trying to solve this question but not able to find any helpful material. It involves factorial with multiplications, $$\frac{8!}{5!}\cdot \frac{7!}{7!10!}$$ I tried crossing 8 and 5 and 7 with ...
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2answers
406 views

Inequality with cube roots

$$(\sqrt{n}+1)^{1/3}-(\sqrt{n}-2)^{1/3} \geq (\sqrt{n+1}+1)^{1/3}-(\sqrt{n+1}-2)^{1/3}$$ $$n\in \mathbb{N}$$ I come upon this inequality when trying to use prove series declension for the Leibnit'z ...
3
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3answers
121 views

Find $a,b \in \mathbb{Z}^+$ such that : $\frac{a^{2}-2}{ab+2}\in \mathbb{Z}$

$1$. Find $a;b\in \mathbb{Z}^+$ such that : $\frac{a^{2}-2}{ab+2}\in \mathbb{Z}$ $2$. Find $m;n>1$ such that : $2^m+3^n=k^2$ $(k\in \mathbb{Z})$ Problem 1. I thought : ...
1
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1answer
82 views

The arithmetic mean of $X$ when arithmetic mean of $X^2 = 29$.

Sorry if my question is a beginner because of my mathematical knowledge is low. arithmetic mean is : $$ \overline x=\dfrac{x_1+x_2+\cdots+x_n}n $$ What method can solve it? $$ ...
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2answers
1k views

Addition and subtraction with exponents

I'm doing an Advanced Functions course right now, and I'm wondering about something. Look at this here evaluation/simplification that I did: http://puu.sh/5w3XQ.png What I'm wondering is about the ...
7
votes
4answers
452 views

How to solve the equations $\sqrt{x-3}+\sqrt{y-3}=\sqrt{y-12}+\sqrt{z-12}=\sqrt{z-27}+\sqrt{x-27}=12$

Let $x,y,z\in R$, and $$\begin{cases} \sqrt{x-3}+\sqrt{y-3}=12\\ \sqrt{y-12}+\sqrt{z-12}=12\\ \sqrt{z-27}+\sqrt{x-27}=12 \end{cases}$$ Find the $x,y,z$. My try: I want use The geometry to ...
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2answers
115 views

we need to find $m+n$.

I just dont understand this question, could any one tell me how to solve this one? A pen costs $13$ dollar and a notebook costs $35$ dollar, let $m$ be the maximum number of items that can be ...
0
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1answer
32 views

An inequality involving quadratic integers

Let $d$ be a positive integer which is not the square of any integer, $x,y \in \mathbb Z$ and $u:=x+y \sqrt d$ $u$ is such that $$ u \geq 1 \;\text{and} \; |u \overline u|=|(x+y \sqrt d)(x-y \sqrt ...
8
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0answers
297 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& ...
12
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2answers
179 views

A problem in fractions from a very old arithmetic textbook

Similar in vein to a problem I posted before here, I would be interested if anyone can give me any pointers as to how one might solve this question from the same arithmetic textbook: "Simplify ...
1
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2answers
75 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x ...
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4answers
49 views

how do we prove $p|q\cdot r \rightarrow p=q$ or $p=r$ (all primes)?

I know this is one of the most fundamental basis of arithmetic but I can't find the result by myself. how do we prove $p|q\cdot r\rightarrow p=q$ or $p=r$? ($p, q, r$ being prime numbers)
1
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1answer
210 views

Evaluating $\int\,\cos(x)\cos(\omega x)\,dx$ using trigonometric addition formulas

I'm looking to solve the integral $$\frac{1}{\pi}\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\, \cos(x)\cos(\omega x)\,dx$$ by rewriting the terms using the trigonometric addition formulaes. It should end ...
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3answers
307 views

What's the difference between arithmetic mean and average?

I'm trying to intuitively understand an average / arithmetic mean: Here's my attempt: In front of me, I see 1 thermos, two computer mice, two pens, and an iPhone. If I sum those, I get ...
1
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2answers
193 views

How can you prove that a value raised to $\frac{1}{n}$ is the n'th root of $x$?

I know that if you raise a value to $\frac{1}{2}$ for example, you take the square root, but that is not what I am asking, what I am asking is; what are you actually doing when raising a value to ...
2
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1answer
1k views

Prove by induction that $a-b|a^n-b^n$ [duplicate]

Given $a,b,n \in \mathbb N$, prove that $a-b|a^n-b^n$. I think about induction. The assertion is obviously true for $n=1$. If I assume that assertive is true for a given $k \in \mathbb N$, i.e.: ...
5
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1answer
122 views

Determine if the number $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ is rational

$ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ I have tried to raising it to the square, but I can't obtain the result. $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - ...
0
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1answer
119 views

What are the smallest possible theories?

Im wondering how we could define a general form for the smallest possible theory in some formal language. In other words, if we have the formal language of first order logic, what is the smallest set ...
2
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0answers
147 views

Convention on the order of scalar multiplication (Multiplier vs multiplicand)

Is there a convention on the order of scalar multiplication? I know there were questions before mine, but I would like to know if such distinction is culturally dependent. This came from a news in ...
2
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1answer
302 views

Arithmetic, Geometric, Harmonic Means

There are situations in which just one of the arithmetic, geometric, and harmonic means are appropriate to use an the other two are meaningless. Is there any situation which more than one of these ...
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2answers
621 views

Come up with some fun “equation Limericks”

We were discussing "Limericks" in my Calculus class. Specifically, "equation Limericks". A Limerick is a poem with five lines. The first, second, and fifth lines should have nine syllables each and ...
2
votes
1answer
86 views

How to prove that $x \gt 0, y \gt 0 \Rightarrow xy \gt 0$

Show that, for every $x,y \in \mathbb Z$, we have: $x \gt 0, y \gt 0 \Rightarrow xy \gt 0$ I've tried this way: supose $x= \overline {(a,b)}, y= \overline {(c,d)}$. Then, $x \gt 0 \Rightarrow ...
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3answers
150 views

Is this a multiplication?

6x-13+4(-3)x=9+2x I'm like, really dumb, I can't tell if if the 4(-3) bit is a multiplication
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3answers
195 views

Multiplication, What is It?

What is multiplication? Upon review logarithms, and square roots, I realized that I have no intuitive grasp of multiplication-well no more so than I have for addition. Is it simply another thing we ...
1
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1answer
56 views

Two variable integer equation

I have the following equation: $$ p^q(2^{q-1}-1)=9p^7q $$ I need to solve for $p$ and $q$. $p$ and $q$ are integers. I think I could take the case $p=0$ separately and for that one $q$ could be ...
0
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1answer
41 views

The sum of the discount is the discount of the sum

Suppose I have a till. I have a special that I give a 10 percent discount on all dinner meals. I want to know that if I calculate the discount on each dinner item and round up the discount and apply ...
2
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1answer
58 views

Arithmetic series relationship with difference of two consecutive cubes. Is this a thing?

Excuse my dodgy notation and my write-up in general, this is the first proof I've done since leaving school a while back. Anywho, has anyone come across anything like this before? Read the whole ...
1
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1answer
214 views

Simplification of $\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}}$

I'm having trouble understanding how this expression: $$\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}} \cdot ...
1
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2answers
85 views

Find $\lim _{n\rightarrow \infty }\dfrac {na^{2}_{n}+1}{\sum ^{n}_{k=1}\left( 1+2+3+\ldots +k\right) }$

Define $\left\{ a_{n}\right\} $ is an arithmetic sequence that all terms are positive integers. If $a_{10}-a_{1}=225$, find $\lim _{n\rightarrow \infty }\dfrac {na^{2}_{n}+1}{\sum ^{n}_{k=1}\left( ...
1
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0answers
60 views

Divisors and cyclotomic polynomials

Let $n \in \mathbb{N}^{\ast}$ and $\Phi_{n}(X)$ be the $n$-th cyclotomic polynomial defined by : $$ \Phi_{n}(X) = \prod \limits_{\substack{1 \leq k \leq n-1 \\ \gcd(k,n)=1}} \Big( X - \exp \big( ...
0
votes
1answer
102 views

If $(a+b)(b+c)(c+a)=2$, then $(a^2+bc)(b^2+ca)(c^2+ab)\leq 1$ [closed]

Prove that if $a,b$ and $c$ are non negative real numbers such that $(a+b)(b+c)(c+a)=2$, then we have $$(a^2+bc)(b^2+ca)(c^2+ab)\leq 1$$
2
votes
3answers
1k views

What is the square root of 1 cm

If $1$cm = $.01$m then shouldn't the square root of $1$cm = the square root of $.01$m but the square root of $1$ = $1$ while the square root of $0.01$ = $0.1$ So my dilemma, is the square root of $1$ ...
0
votes
2answers
213 views

Help with limit of radical expression

$$\lim_{x \to \infty} (\sqrt{x^2-49}-\sqrt{x^2-16} ) $$ I multiplied by the conjugate radical expression: $$=(\sqrt{x^2-49}-\sqrt{x^2-16}) \times (\sqrt{x^2-49}+\sqrt{x^2-16}) $$ $$= ...