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
4answers
945 views

Not quite Fermat's Last Theorem

Prove that the equation $n^a + n^b = n^c$, with $a,b,c,n$ positive integers, has infinite solutions if $n=2$, and no solution if $n\ge3$.
17
votes
3answers
1k views

What skill do I lack to factor multivariate polynomials?

Ok so I can factor easily regular quadratic polynomials, i.e. $5x^2+7x+9$ (I'm not sure whether that's prime, just made it up), and I was working on solving $y^2+(x^2+2x−2)y+(x^3−x^2−2x)$ by ...
17
votes
4answers
1k views

Why do all parabolas have an axis of symmetry?

And if that's just part of the definition of a parabola, I guess my question becomes why is the graph of any quadratic a parabola? My attempt at explaining: The way I understand it after some ...
17
votes
3answers
481 views

When are algebraic expressions equivalent?

This question arose when I was going to determine the domain for $f \circ f(x)$. Let $f(x) = \dfrac{1-x}{1+x}$. $f \circ f(x) = x, \quad$ But the domain is not $\mathbb{R}$ because $f(x)$ is undefined ...
17
votes
2answers
21k views

Weird E letter? (sigma) [duplicate]

Possible Duplicate: What does the math notation $\sum$ mean? My school's prescribed book uses the weird letter E character without explaining what it is in the first chapter when it talks ...
17
votes
2answers
245 views

square root / factor problem $(A/B)^{13} - (B/A)^{13}$

Let $A=\sqrt{13+\sqrt{1}}+\sqrt{13+\sqrt{2}}+\sqrt{13+\sqrt{3}}+\cdots+\sqrt{13+\sqrt{168}}$ and $B=\sqrt{13-\sqrt{1}}+\sqrt{13-\sqrt{2}}+\sqrt{13-\sqrt{3}}+\cdots+\sqrt{13-\sqrt{168}}$. Evaluate ...
17
votes
2answers
292 views

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
17
votes
2answers
370 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
90k 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
2answers
490 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. ...
17
votes
1answer
262 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
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.
16
votes
7answers
711 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
120k 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
284 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
513 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
453 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
217 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
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 ...
16
votes
2answers
419 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
3answers
2k views

How to solve equations to the fourth power?

Is it possible to manually retrieve the value of $y$ from the following equation $$153y^2-y^4=1296$$ WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved? What I've achieved ...
15
votes
7answers
856 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
2answers
3k views

Integration by partial fractions; how and why does it work?

Could someone take me through the steps of decomposing $$\frac{2x^2+11x}{x^2+11x+30}$$ into partial fractions? More generally, how does one use partial fractions to compute integrals ...
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
4answers
765 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
780 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
760 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
4k 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
841 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
419 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
708 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
7answers
749 views

Summation Theorem how to get formula for exponent greater than 3

I'm studying in the summer for calculus 2 in the fall and I'm reading about summation. I'm given these formulas: \begin{align*} \sum_{i=1}^n 1 &= n, \\ \sum_{i=1}^n i &= \frac{n(n+1)}{2},\\ ...
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
340 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
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
313 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
275 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
568 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
706 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
410 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
4k 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 $b^2 + c^2 \geq 2bc~$. ...