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

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3answers
95 views

Evaluate $\sqrt{2x} \cdot \sqrt x$

So I have this problem: $\sqrt{2x} \cdot \sqrt x =\ldots$ I already have the answer which is $x\sqrt2$, but I just can't understand it. Someone that could help?
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4answers
66 views

How can I determine the function for summing a series of integers? [duplicate]

I want to sum a series of integers, say from 1 to 5, but I want to generalise this function. I have seen that the function is: $$ \text{sum} = \frac{n(n+1)}{2} $$ where $n$ is the number to sum up ...
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1answer
18 views

Able to adjust a Function/Formula with Weighted Variables to correct for changes in a variable?

I apologize for the somewhat cryptic title as I don't quite know how to word it. I have a somewhat abstract question that may have a simple answer. But I am wracking my mind all over this! So ...
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2answers
53 views

Division with Remainder

When I divide 48/12 by hand, it's 0 Remainder 12. But when I divide it in a calculator, it's .25. Why is this? How does 0R12 turn into .25? Thanks. ANSWER (system won't let me answer my own ...
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2answers
49 views

Why 3 is a multiplicative inverse of 7 in modular arithmetic?

why is 3 a multiplicative inverse of 7 in modular arithmetic of 5 ? I'm not able to understand how this is true. PS: I know 3*7-1 % 5 = 0. I'm not able to make sense of inverses in modular ...
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1answer
60 views

Simplify $\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$

I am trying to find the value of: $$\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$$ I have the key with the answer $\sqrt 5$ but am wondering how I can easily get to that ...
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1answer
44 views

Change of the average when a number is removed

The average of 21 members is 30. The largest number is 50. If we remove the largest number then the average of remaining numbers will be?
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1answer
38 views

Two conversions to base three yield different results

How are there two different conversion results for the same bases? Am I doing something wrong?
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0answers
32 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
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2answers
85 views

How prove that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $

How check that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $?
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1answer
51 views

Factorization of rational powers of rational numbers

If I am not wrong, rational powers of rational numbers can be factorized in an unique way as product of rational powers of different prime numbers: $10^{1/2} = 2^{1/2} \cdot 5^{1/2}$ $(8/9)^{1/6} = ...
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3answers
84 views

Multiplication of the numbers $1$ to $n$

Let's say I want to find the product of $1,2,3, \dots, 10$. Do I need to do $1 \cdot 2 \cdot 3 \cdot \dots \cdot 10$ manually or is there an easier way to do it? Something like the sumation of $1$ ...
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0answers
92 views

Given $m^k\le n <m^{k+1}$ find $x$ and $y$ such that $x\cdot m^k+y=n$

Let $n,m,k\in\mathbb{N}$. Assume $m^k\le n <m^{k+1}$. Find $x,y\in\mathbb{N}$ such that (1) $x\cdot m^k+y=n$ (2) $0<x<m$ (3) $0\le y<m^k$ My question: does there exist a general ...
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1answer
83 views

About translating subsets of $\Bbb Z.$

This is a continuation of About translating subsets of R2. Is it possible to find a pair of sets $A,B\subseteq\Bbb Z$ such that A is a union of translated (only translations are allowed) copies of ...
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4answers
167 views

Diophantine equation abc + abd + acd + bcd= 1

Is there a reference which classifies or at least gives an infinite family of integer solutions to the above equation? A solution to the problem would also be great obviously.
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2answers
56 views

Simplifying Surd Fractions

can someone show me how to simple surd fractions such as: $$\frac{{8\sqrt 3 }}{2}$$ Can someone please help me here?
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1answer
37 views

Help with finding the arithmetic mean of all the radii from the center to the edge of an ellipse?

So far I approached this problem computationally, I decided to take all the radii add them up, by distance formula, then divide by the number of radaii. To make the distribution even, I rotated the ...
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4answers
55 views

How to work out the percentage?

I'm not exactly sure how to do this. I know the answer is $£64,000$ but whatever I try, nothing is working. i.e. $£80,000$ x $25%$ $= 2,000,000$ devided by $100$ $=$ $20,000$ $£80,000 - £20,000 = ...
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1answer
56 views

How is this definition of a constant divided by zero called?

I divide a constant by zero. One example is the following: 2/0 My father told me he learned at school earlier that the result is "not defined". If I enter this arithmetic problem in Wolfram Alpha, I ...
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3answers
164 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
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1answer
38 views

Euler product of Dirichlet series

Let $f$ be an arithmetic function such $f(n_1n_2)=f(n_1)f(n_2)$ for all $n_1,n_2 \in \mathbb{N}$ with $\gcd(n_1,n_2)=1$. Suppose we know that the Dirichlet series $$F(s) = \sum_{n=1}^{\infty}f ...
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1answer
26 views

How to calculate component amounts so their individual additives equal 3%

I have a list of chemical formulas that are each comprised of a number of base components. Two of the base components contain an additive, $X$. This additive needs to exist in each formula at a ...
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2answers
74 views

How to turn arbitrary fractions into arbitrary egyptian fractions?

I am reading Stillwell's Numbers and Geometry. There is an exercise about Egyptian fractions which is the following: I've tried to do it in the following way - Expressing an arbitrary fraction ...
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3answers
133 views

How addition and multiplication works

Lets say i am doing 12 + 13 by using the addition method that we know. i mean first we write 13 below 12, then we do 2+3 and then 1+1. The result can be validated as 25 (or true) by doing the counting ...
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1answer
44 views

Why aren't my % Changes additive?

I'm struggling conceptually with the fact that I have a variable C that is the product of 2 other variables, A and B yet the annual change as a % in C is not the annual change % of A + B. e.g As ...
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3answers
52 views

When a fraction is raised to a negative exponent, do you normally transform it to 1 over the fraction, or invert the fraction?

My text shows that $$\left(\frac{3a^2}{4b}\right)^{-3}=\frac{1}{\left(\frac{3a^2}{4b}\right)^{3}}.$$ It also shows that $$\frac{1}{\frac{144}{b}}=\frac{b}{144}.$$ In the first equation, it seems ...
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2answers
88 views

Why does $(3^{1/2})(10^{1/2})=30^{1/2}$ but $(3a^2)(10a^2)=30a^4$?

$(3a^2)(10a^2)=30a^4$? In that equation the exponents are added. Why does $(3^{1/2})(10^{1/2})=30^{1/2}$. In that equation the exponents are not added. Why?
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2answers
49 views

Square root each term (clarification on polynomials?)

So I'm in Algebra 2, and right now we're learning about conic sections (circles/ellipse/etc). I thought some problems in the workbook looked weird, like this one: ...
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1answer
37 views

What do you call this arithmetical property and how do you prove it?

I think I can prove this property only when exponents are integers. But here is my example: $a(x-b)^{1/2} = (a^2x-a^2b)^{1/2}$ This type of foiling is weird and I wish to see a proof of this. ...
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2answers
48 views

Gambling question: multiply quotes.

Reading various betting forum I came across different threads claiming betting multiple is worse than betting on single events. Could you explain why? [Clairification for the ones not familiar with ...
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2answers
99 views

What is the value of $(72^2 - 64^2) : (44^2 - 24^2)$ [closed]

What is the value of $(72^2 - 64^2) : (44^2 - 24^2)$ How to calculate this without calculator?
3
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1answer
93 views

An operation with respect to which the set of prime numbers is closed

Like every (semi-)decidable set of natural numbers the set $P$ of prime numbers is diophantine, i.e. there are two polynomials $p(x)$, $q$ with natural coefficients and exponents – the first of ...
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2answers
177 views

Combination of quadratic and arithmetic series

Problem: Calculate $\dfrac{1^2+2^2+3^2+4^2+\cdots+23333330^2}{1+2+3+4+\cdots+23333330}$. Attempt: I know the denominator is arithmetic series and equals ...
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1answer
12 views

Find the first $3$ terms of the two possible geometric progressions.

The fourth term of a G.P is $3$ and the sixth term is $147$. Find the first $3$ terms of the two possible geometric progressions. Can you help me find $a$ and $r$? It is too complicated. I took two ...
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1answer
25 views

Find the sum of the 10th and 11th terms of the G.P.

The third term of a geometric progression of positive terms is $\frac{6}{25}$ and the seventh term is $1\frac{23}{27}$. Find the sum of the 10th and 11th terms of the G.P., giving your answers correct ...
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0answers
135 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
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2answers
47 views

How many wins does my team have to get?

I am watching a tournament. 16 teams are participating in the playoff. Every team play against another once. If you win, you score 1 point, if you lose you dont score any point (the outcome of a match ...
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3answers
67 views

Square root of a squared number changes sign, which to apply first?

Heres something Ive always found interesting. Supose we have a variable $x$, and $x$ equals a negative number: Say: $$x=-17$$ Now, I can apply a square to both sides of the equation and preserve ...
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2answers
26 views

Profit, Loss and percentage problem.

A dishonest businessman professes to sell his articles at cost price but he uses false weight by which he cheats by 10% while buying and 10% while selling. find his profit percentage?
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1answer
53 views

How many answers can be created using the elementary arithmetic operators?

If I gave you an amount of $n$ numbers, how many anwswer will you be able to create using the elementary arithmetic operators ($+, -, \times, /$)? These are the rules: All numbers ...
2
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2answers
58 views

Geometrical proof of the existence of square roots

This is quite an easy question, but it's been troubling me and I can't manage to work it out. I've been reading the book A Concise Introduction to Pure Mathematics (M. Liebeck), so I'll quote the ...
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1answer
19 views

Square Root in inequalities

I thought I could be able to find an explanation of this (and perhaps I knew it a long time ago) but I cannot find an answer now. If I assume that $b^2 \leq |x|$, then is it true that both $b \leq ...
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1answer
50 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
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2answers
40 views

How to calculate price for stamps?

I'm trying to create a programm that has the ability to calculate the most effective way of breaking down the number of stamps (3 and 5 cent) necessary for a consignment. That is - if a transmission ...
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1answer
30 views

Highest Common Factor to Infinity

Imagine you have a set of integers of x.For example: 7 9 11 13 Let us imagine that y is 1. Then for each nth generation you added 1 to each member of the set, found the HCF of the set and set y to ...
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2answers
43 views

Quarters weigh 6 grams while dimes weigh 2 grams.

Quarters weigh $6$ grams while dimes weigh $2$ grams. Tiffany has $\$5.35$ worth of quarters and dimes in her pocket weighing a total of $124$ grams. How many quarters does Tiffany have?
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2answers
278 views

Prove that $5$ does not divide $52$

If I suppose that $5$ divides $52$, then there would exist an $ s \in \mathbb Z $ such that $ 5s = 52 $. There is no such s, because $5(10) = 50$, and $5(11) = 55$. I'm not convinced with this proof, ...
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1answer
28 views

Dumb question about the division algorithm.

The theorem about the division algorithm says: Given a, b $ \in \mathbb{Z}, b \neq{0}, $ there exist unique numbers q and r , $q,r \in \mathbb Z $such that $ a = bq + r , 0 \lt r \lt |b| $. Can q ...
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5answers
183 views

Prove that for all real numbers $x$ and $y$, if $x+y \geq 100$, then $x \geq 50$ or $y \geq 50$.

I'm confused about the following question in my math textbook. Prove that for all real numbers $x$ and $y$, if $x+y \geq 100$, then $x \geq 50$ or $y \geq 50.$ The or is what gets me. For or to be ...
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1answer
18 views

Commpossed percentage over same base

Maybe this sounds a Little trivial. I have a value and Over this value apply percentage (a), then over this result apply other percentage (b) Could say : Value = 145760 a= 8% b= 40% My actual ...