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Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
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Sets of integers in which every element is the sum of two others

i) Does there exists a nonempty subset of integers $S$, such that for all $a \in S$, there exists $b,c \in S$ such that $a = b + c$, where $a, b, c$ are distinct integers. Edited to add: ii) Can an $S$ with these properties be finite, and also have…
AgCl
  • 6,292
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votes
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Solution by radicals of $(1+x)^n=x^m$

Given the equation $$(1+x)^n=x^m$$ where $m$ and $n$ are two different natural numbers, I was trying to find as many solution as possible expressing them without transcendent functions. WLOG we can suppose $n
N74
  • 2,479
11
votes
4 answers

Place the numbers by their size.

Place the following numbers by their size: $$A=2^{4^{2^{.^{.^{.^{2^{4}}}}}}},B=4^{2^{4^{.^{.^{.{4^{2}}}}}}},C=2^{2^{2^{.^{.^{.^{2^{2}}}}}}}$$ In number $C$ there are $2000$ "$2$" digits, and in numbers $B,A$ there are $500$ "$2$" and $500$ "$4$"…
Taha Akbari
  • 3,559
11
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9 answers

Why is it unacceptable to say "the range is a function of the domain"?

I understand that a function is defined as a correspondence between two sets, the domain and the range. While the definitions of the domain of a function and the range of a function are said to be contained within the definition of a function. Now,…
user685252
  • 123
11
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2 answers

A formula to convert a counter-clockwise angle to clockwise angle with an offset

I have an angle in the coordinate system where $0^\circ$ is East, $90^\circ$ is North, $180^\circ$ is West, and $270^\circ$ is South. I need to convert them to this one, where $0^\circ$ is North, $90^\circ$ is East, $180^\circ$ is South, and…
Andrei Glingeanu
  • 255
11
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7 answers

Prove: $(a + b)^{n} \geq a^{n} + b^{n}$

Struggling with yet another proof: Prove that, for any positive integer $n: (a + b)^n \geq a^n + b^n$ for all $a, b > 0:$ I wasted $3$ pages of notebook paper on this problem, and I'm getting nowhere slowly. So I need some hints. $1.$ What…
Matt
  • 441
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1 answer

Solving for real x and y

If x and y be real numbers such that, $x^3 - 3x^2 + 5x = 1$ and $y^3 - 3x^2 + 5y = 5$; Find $(x + y)$ From an old Russian olympiad. I tried to make the equations homogenous by substituting for $1 = x^3 - 3x^2 + 5x$ in the second equation for $5 *…
buzaku
  • 589
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2 answers

Help writing proof for $\sqrt{a^2 + b^2} \neq \sqrt[3]{a^3 + b^3}$

Like the title says I'm attempting to write a existence proof showing "that there exists no non-zero real numbers a and b such that $\sqrt{a^2 + b^2} = \sqrt[3]{a^3 + b^3}$". I'm having trouble finding a starting point. I've tried manipulating the…
AvatarOfChronos
  • 1,701
10
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5 answers

Why add before dividing in this equation?

For the following equation, I know the correct answer is $9$: $$ x / 3 + 2 = 5 $$ You subtract $2$ from each side, and the multiply each side by $3$... But why do you subtract the $2$ first? Doesn't the order of operations say I should do division…
user138021
  • 115
10
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2 answers

Order of operations involving mod (%)

I was taking a programming test last night that had a math equation that simplified to 11 % 2 * 3, no () or likewise. When I compute it, being taught modulous occurs at the same level of multiplication or division. As a result I get 11 % 2 * 3 1 *…
traisjames
  • 201
10
votes
4 answers

Can you prove $(x-y)^3+(y-z)^3+(z-x)^3 = 3(x-y)(y-z)(z-x)$?

Show that $$(x-y)^3+(y-z)^3+(z-x)^3 = 3(x-y)(y-z)(z-x)$$ This can be shown through expansion but there is a more elegant solution I cannot discover anything I would consider elegant. Can anyone help?
user2321
  • 699
10
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7 answers

If $a+\frac1b=b+\frac1c=c+\frac1a$ for distinct $a$, $b$, $c$, how to find the value of $abc$?

If $a, b, c$ be distinct reals such that $$a + \frac1b = b + \frac1c = c + \frac1a$$ how do I find the value of $abc$? The answer says $1$, but I am not sure how to derive it.
Quixotic
  • 22,431
10
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5 answers

How to solve equations of this form: $x^x = n$?

How would I go about solving equations of this form: $$ x^x = n $$ for values of n that do not have obvious solutions through factoring, such as $27$ ($3^3$) or $256$ ($4^4$). For instance, how would I solve for x in this equation: $$x^x = 7$$ I am…
John
  • 113
10
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2 answers

A problem regarding rational and irrational numbers.

Let $n$ be a positive integer greater or equal to $2$. Prove that there are infinitely many irrational numbers $a$ such that $a+a^2+\cdots+a^n$ is rational. Well, let $p$ be a prime. We consider the equation: $x^n + x^{n-1} +\cdots+ x=1/p$, which…
user85046
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