Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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17
votes
2answers
368 views

Rigorous proof that $\displaystyle\frac{1}{3} = 0.333\ldots$

I'm a PreCalculus student trying to find a rigorous proof that $\displaystyle\frac{1}{3} = 0.333\ldots$, but I couldn't find it. I think (just think) that this proof would start by proving that ...
17
votes
1answer
86k views

Solving Triangles (finding missing sides/angles given 3 sides/angles)

What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
17
votes
1answer
3k views

Strange old multiplication table

Today I read an article about chalk boards from 1917 discovered in an Oklahoma school. One of the chalkboards included the following curious image: (Oklahoma City Public Schools) The article ...
17
votes
1answer
261 views

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes. I've been thinking about this question, but ...
16
votes
13answers
1k views

$xy=1 \implies $minimum $x+y=$?

If $x,y$ are real positive numbers such that $xy=1$, how can I find the minimum for $x+y$?
16
votes
7answers
709 views

A geometric look at $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$?

Is there a geometric way of looking at the relationship between the positive real numbers $a$, $b$ and $c$ if $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$?
16
votes
3answers
114k views

How to Determine if a Function is One-to-One

I am looking for the "best" way to determine whether a function is one-to-one, either algebraically or with calculus. I know a common, yet arguably unreliable method for determining this answer would ...
16
votes
2answers
1k views

Interesting Question on Ants

A horizontal stick is one metre long. Fifty ants are placed in random positions on the stick, pointing in random directions. The ants crawl head first along the stick, moving at one metre per minute. ...
16
votes
3answers
277 views

Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

My idea is that the two functions are not the same since for the first function, the domain of the function is only non negative reals for the numerator and positive reals for the denominator. ...
16
votes
6answers
505 views

$\log_9 71$ or $\log_8 61$

I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
16
votes
5answers
448 views

Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$

How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$ Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't ...
16
votes
2answers
6k views

Strategies to denest nested radicals.

I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into ...
16
votes
3answers
213 views

How to prove $\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$

I was given a challenge of sorting the following numbers. $\Large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$. After some work I was able to figure out the order. How can one ...
16
votes
2answers
410 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
15
votes
7answers
850 views

How to calculate $z^4 + \frac1{z^4}$ if $z^2 + z + 1 = 0$?

Given that $z^2 + z + 1 = 0$ where $z$ is a complex number, how do I proceed in calculating $z^4 + \dfrac1{z^4}$? Calculating the complex roots and then the result could be an answer I suppose, but ...
15
votes
12answers
2k views

How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations

I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong? $$(a-b)^3 + (b-c)^3 + (c-a)^3 ...
15
votes
6answers
2k views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
15
votes
4answers
759 views

Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$

Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid? I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
15
votes
4answers
777 views

Proving that $\sum\limits_{i=1}^k i! \ne n^2$ for any $n$ [duplicate]

Possible Duplicate: How to prove that the number 1!+2!+3!+…+n! is never square? Show that $\displaystyle\sum\limits_{i=1}^k i!$ is never a perfect square for $k\ge4$ I could prove ...
15
votes
6answers
753 views

Solution to $1-f(x) = f(-x)$

Can we find $f(x)$ given that $1-f(x) = f(-x)$ for all real $x$? I start by rearranging to: $f(-x) + f(x) = 1$. I can find an example such as $f(x) = |x|$ that works for some values of $x$, but not ...
15
votes
6answers
3k views

No radical in the denominator — why? [duplicate]

Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to ...
15
votes
4answers
814 views

Factor $x^4 - 11x^2y^2 + y^4$

This is an exercise from Schaum's Outline of Precalculus. It doesn't give a worked solution, just the answer. The question is: Factor $x^4 - 11x^2y^2 + y^4$ The answer is: $(x^2 - 3xy -y^2)(x^2 + ...
15
votes
4answers
417 views

If $x$ and $y$ are rational numbers and $x^5+y^5=2x^2y^2,$ then $1-xy$ is a perfect square.

Prove that if $x, y$ are rational numbers and $$ x^5 +y^5 = 2x^2y^2$$ then $1-xy$ is a perfect square.
15
votes
4answers
1k views

How many solutions for this equation?

$$ \frac{x-4}{(x-1)} = \frac{1-4}{(x-1)} $$ Can someone tell me how many solutions are there for the above equation? MY APPROACH: I cross multiplied the equations and re-arranged to get a quadratic ...
15
votes
2answers
650 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
15
votes
3answers
2k views

Riddle with Pi = 3

This is a riddle someone posted on Google+, so please forgive it's triviality - I'm asking here because I just can't figure out what exactly is wrong, and it really bugs me ;) I think something is ...
15
votes
2answers
2k views

Significance of $\displaystyle\sqrt[n]{a^n} $?

There is a formula given in my module: $$ \sqrt[n]{a^n} = a \text{ if $n$ is odd } $$ $$ \sqrt[n]{a^n} = |a| \text{ if $n$ is even } $$ I don't really understand the differences between them, ...
15
votes
3answers
330 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
15
votes
5answers
9k views

Good book for high school algebra

I'm gonna take a Calculus course next year, my professor suggest me to review high school algebra. I want to know, which book is good for refresh knowledge on high school algebra?
15
votes
2answers
11k views

How to prove a limit exists using the $\epsilon$-$\delta$ definition of a limit

I understand how to find a limit. I understand the concept of the $\epsilon$-$\delta$ definition of a limit. Can you walk me through what we're doing in this worked example? It is from my student ...
15
votes
1answer
1k views

Nested Square Roots

How would one go about computing the value of $X$, where $X=5^0+ \sqrt{5^1+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+\sqrt{5^{16}+\sqrt{5^{32}+\dots}}}}}}$ I have tried the standard way of squaring then trying ...
15
votes
2answers
3k views

Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...
15
votes
8answers
290 views

What is the connection between the discriminant of a quadratic and the distance formula?

The $x$-coordinate of the center of a parabola $ax^2 + bx + c$ is $$-\frac{b}{2a}$$ If we look at the quadratic formula $$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ we can see that it specifies two ...
15
votes
1answer
316 views

Solve $x^x=2x$ where $x\in\mathbb C$.

Solve $x^x=2x$ where $x\in\mathbb C$. Obviously, one solution is $x=2$. By WA, another solution is $x=0.346...$. How to solve it analytically, e.g. using Lambert W function? Thank you.
15
votes
2answers
391 views

High school algebra textbooks for gifted students

Cross-posted to Math Educators Stack Exchange. (link) I am looking for high school algebra/mathematics textbooks targeted at talented students, as preparation for fully rigorous calculus à la Spivak. ...
15
votes
2answers
272 views

How much math do we need to prove all simple numeric identities?

Consider real numeric expressions build only from integers, operators $+,-,\times,/$ and taking a positive expression to a power (no variables involved), e.g. $$\frac{2}{7},\ 2^{1/2},\ ...
14
votes
8answers
1k views

Variation on Pythagoras: If $a^2 + b^2 = c^2$, then $a + b \leq c\sqrt{2}$

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with derivative of the Pythagorean Theorem using calculus, trigonometry, ...
14
votes
3answers
563 views

Is there any “superlogarithm” or something to solve $x^x$? [duplicate]

Is there any "superlogarithm" or something to solve an equation like this: $$x^x = 10?$$
14
votes
7answers
704 views

$n^5-n$ is divisible by $10$?

I was trying to prove this, and I realized that this is essentially a statement that $n^5$ has the same last digit as $n$, and to prove this it is sufficient to calculate $n^5$ for $0-9$ and see that ...
14
votes
3answers
404 views

$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$

How can I find the following product using elementary trigonometry? $$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$ I have tried using a substitution, but nothing ...
14
votes
5answers
3k views

Inequality: $(x + y + z)^3 \geq 27 xyz$

Edit: $a,b,c$ and $x,y,z$ are positive, real numbers. Since $(a-b)^2 \geq 0~$, $a^2 + b^2 - 2ab\geq0~$ and $a^2 + b^2 \geq 2ab~$. Similarly, $a^2 + c^2 \geq 2ac~$ and $b^2 + c^2 \geq 2bc~$. ...
14
votes
4answers
629 views

Process to show that $\sqrt 2+\sqrt[3] 3$ is irrational

How can I prove that the sum $\sqrt 2+\sqrt[3] 3$ is an irrational number ??
14
votes
4answers
573 views

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
14
votes
3answers
564 views

An incorrect method to sum the first $n$ squares which nevertheless works

Start with the identity $\sum_{i=1}^n i^3 = \left( \sum_{i = 1}^n i \right)^2 = \left(\frac{n(n+1)}{2}\right)^2$. Differentiate the left-most term with respect to $i$ to get $\frac{d}{di} ...
14
votes
3answers
647 views

Evaluate $ \displaystyle \lim_{x\to 0}\Bigg( \frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}\Bigg) $

Evaluate $$ \displaystyle \lim_{x\to 0}\Bigg( \frac {(\cos(x))^{\sin(x)} - \sqrt{1 - x^3}}{x^6}\Bigg) $$ I tried to use L'Hospital rule but it got very messy. Moreover I also tried to analyze ...
14
votes
4answers
343 views

Show the identity $\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}=-\frac{a-b}{a+b}\cdot\frac{b-c}{b+c}\cdot\frac{c-a}{c+a}$

I was solving an exercise, so I realized that the one easiest way to do it is using a "weird", but nice identity below. I've tried to found out it on internet but I've founded nothingness, and I ...
14
votes
2answers
2k views

What is the best way to solve an equation involving multiple absolute values?

An absolute value expression such as $|ax-b|$ can be rewritten in two cases as $|ax-b|=\begin{cases} ax-b & \text{ if } x\ge \frac{b}{a} \\ b-ax & \text{ if } x< \frac{b}{a} \end{cases}$, ...
14
votes
2answers
380 views

how to compare $\sin(19^{2013}) $ and $\cos(19^{2013})$

how to compare $ \sin(19^{2013})$ and $\cos (19^{2013})$ or even find their value range with normal calculator? I can take $2\pi k= 19^{2013} \to \ln(k)= 2013 \ln(19)- \ln(2 \pi)=5925.32 \to k= ...
14
votes
2answers
490 views

proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$ [duplicate]

i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows: $$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
14
votes
3answers
181 views

Finding the real solutions to $16^{x^{2} + y } + 16^{y^{2}+ x} = 1$

We have , $16^{x^{2} + y } + 16^{y^{2}+ x} = 1$ , then we have to find all the real values of $x$ and $y$.I tried this question but i am not able to proceed because I am not able to simplify this ...