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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12
votes
2answers
324 views

Does $z^i=i^z$ have any solutions, beside $z=i$?

Does this equation have any solutions: $$\large{z^i=i^z}$$ Putting polar form of $z$ is better for LHS, But rectangular form is suitable for RHS ! What to do? Thanks!
12
votes
2answers
493 views

A highschool factoring problem

$x+y+z=0$ $x^3+y^3+z^3=9$ $x^5+y^5+z^5=30$ $xy+yz+zx=?$ I solved this problem by setting $xy+yz+zx=k$ and using the cubic equation with roots $x,y,z$. But is there any other methods?
12
votes
2answers
704 views

My Daughter's 4th grade math question got me thinking

Given a number of 3in squares and 2in squares, how many of each are needed to get a total area of 35 in^2? Through quick trial and error (the method they wanted I believe) you find that you need 3 ...
12
votes
4answers
180 views

Under what conditions does $x^{\frac{b}{c}} = (x^b)^\frac{1}{c}$ hold?

It is very common to use the formula $$x^{\frac{b}{c}} = (x^b)^\frac{1}{c}$$ to simplify the evaluation of a fractional exponent. I want to know what circumstances allow us to do this step. For ...
12
votes
3answers
326 views

On solving $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}=\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+x}}}}$

How do we show that there is only one solution to,$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}=\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+x}}}}$$ I guess it is only $x=2$. Please help.
12
votes
5answers
5k 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?
12
votes
2answers
412 views

Number of real positive roots of a polynomial?

Consider the polynomial $$f(x)=x((1+x^n)^n+a^n)-a(1+x^n)^n,$$ where $n\geq 2$ is a positive integer and $a$ is a positive real number. I'm interesting in deducing the number of positive real roots ...
12
votes
3answers
412 views

Number Theory or Algebra?

Prove that if $4^m-2^m+1$ is a prime number, then all the prime divisors of $m$ are smaller than $5$ I initially thought about putting $4^m-2^m+1=p$ where $p$ is some prime and after eliminating ...
12
votes
1answer
255 views

How to prove $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$

Question: Let $a,b,c>0$ are give numbers and $x>0$, such that $$ \sqrt{\dfrac{a+b+c}{x}}=\sqrt{\dfrac{b+c+x}{a}}+\sqrt{\dfrac{c+a+x}{b}}+\sqrt{\dfrac{a+b+x}{c}} $$ show that $$ ...
12
votes
2answers
99 views

Finding all solutions to the equation $|||||x|-1|-1|-1|-1|=0$

I was presented this question by a student I was tutoring: Suppose $x \in \mathbb{R}$. Find all solutions of the equation $$|||||x|-1|-1|-1|-1|=0.$$ What I explained to the student: Given ...
12
votes
1answer
345 views

When $\sin x, \cos x$ are $\mathbb{Q}$-linear combinations of square roots

Suppose $x\in\Bbb R$ is such that $$\sin x=\sum_{i=1}^m x_i\sqrt{r_i},\quad \cos x=\sum_{j=1}^n y_j\sqrt{s_j}$$ for some $x_i, r_i, y_j, s_j \in\Bbb Q \ , \ |x_i|=|y_j|=1$. Show that ...
12
votes
1answer
167 views

Value of $\sin (2^\circ)\cdot \sin (4^\circ)\cdot \sin (6^\circ)\cdots \sin (90^\circ) $

How can I calculate the value of $\sin (1^\circ)\cdot \sin (3^\circ)\cdot \sin (5^\circ)\cdots \sin (89^\circ)$ $\sin (2^\circ)\cdot \sin (4^\circ)\cdot \sin (6^\circ)\cdots \sin (90^\circ)$ My ...
12
votes
4answers
288 views

Should the domain of a function be inferred?

It is a common practice to have students of elementary algebra infer the domain of a function as an exercise. I believe this is contrary to the spirit of the definition of a function as a collection ...
12
votes
1answer
517 views

Find the sum $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + …+ \frac{1}{\sqrt{99}+\sqrt{100}}$

I would like to check I have this correct Find the sum $$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + ...+ \frac{1}{\sqrt{99}+\sqrt{100}}$$ Hint: rationalise the denominators ...
12
votes
1answer
224 views

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
12
votes
1answer
319 views

Solving a formal power series equation

I want to find a function $f(x,y)$ which can satisfy the following equation, $$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = \exp \left[ \sum _{n=1} ^\infty ...
12
votes
2answers
847 views

Find all solutions of an exponential equation

Find the product of all the solutions of $\displaystyle\left(\frac{x^2-5x}{6}\right)^{x^2-2}=1$ times the number of solutions. I don't know how to solve an exponential equation, so I've done as ...
12
votes
1answer
266 views

If the product of $x$ positive integers is $n!$ What is the smallest possible value their sum can have?

If the product of $x$ positive integers is $n!$ What is the smallest possible value their sum can have? I was wondering what could be the most efficient strategy to solve this problem for ...
11
votes
11answers
2k views

Solve $4^{9x-4} = 3^{9x-4}$

I am having some trouble trying to solve $$4^{9x-4} = 3^{9x-4}$$ I tried to make each the same base but then I'm becoming confused as to what to do next. These are the steps I took: ...
11
votes
10answers
2k views

Find five consecutive odd integers such that their sum is $55$.

So my professor asked us to do an Olympiad exercise which says that the sum of five consecutive odd integers is $55$, find those integers. But I've never seen such an exercise so it is quite new and ...
11
votes
5answers
2k views

Simplifying $\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}$

How do I simplify $\sqrt{(4+2\sqrt{3})}+\sqrt{(4-2\sqrt{3})}$? I've tried to make it $x$ and square both sides but I got something extremely complicated and it didn't look right. I got $2\sqrt{3}$ ...
11
votes
9answers
2k views

7th class question my daughter asked and need answer if possible

My daughter has asked me to solve this question but I am unable to than I thought to post it here may be someone help. Q : The watchman works 4 days a week and has a rest on the fifth day. He had ...
11
votes
8answers
597 views

How to factor quadratic $ax^2+bx+c$?

How do I shorten this? How do I have to think? $$ x^2 + x - 2$$ The answer is $$(x+2)(x-1)$$ I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link to ...
11
votes
7answers
513 views

Can someone show me why this factorization is true?

$$x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \dots + xy^{n-2} + y^{n-1})$$ Can someone perhaps even use long division to show me how this factorization works? I honestly don't see anyway to "memorize ...
11
votes
8answers
11k views

What is a simple example of a limit in the real world?

This morning, I read Wikipedia's informal definition of a limit: Informally, a function f assigns an output $f(x)$ to every input $x$. The function has a limit $L$ at an input $p$ if $f(x)$ is ...
11
votes
7answers
2k views

Factorize the polynomial $x^3+y^3+z^3-3xyz$

I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?
11
votes
4answers
501 views

Simplifying $\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt {5 +\cdots}}}}$

How to simplify the expression: $$\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt{\cdots}}}}.$$ If I could at least know what kind of reference there is that would explain these type of expressions that would be ...
11
votes
5answers
4k views

Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$

Is there any formula for this series? $$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
11
votes
3answers
984 views

Simplification of expressions containing radicals

As an example, consider the polynomial $f(x) = x^3 + x - 2 = (x - 1)(x^2 + x + 2)$ which clearly has a root $x = 1$. But we can also find the roots using Cardano's method, which leads to $$x = ...
11
votes
4answers
1k views

Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$

I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a ...
11
votes
5answers
414 views

How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$

Let $p \in \mathbf{N}$. I don't know how to prove that $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^j=0 \textrm{ for } j \in \{0,\ldots,p-1\},$$ and $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^p=p!$$ ...
11
votes
4answers
6k views

How to remember the trigonometric identities

I have a test tomorrow and I am having trouble remembering those pesky trigonometrical identities (such as $1-\cos x=2\sin^2(\frac{x}{2})$ ) Do you guys have any tips on how I can remember these? ...
11
votes
3answers
759 views

Division by imaginary number

I ran into a problem dividing by imaginary numbers recently. I was trying to simplify: $2 \over i$ I came up with two methods, which produced different results: Method 1: ${2 \over i} = {2i \over ...
11
votes
3answers
400 views

$f(f(x))$ has no fixed points if $f(x)$ has no fixed points

Assume that $f(x)=x$ has no real roots where $$f(x) = ax^2+bx+c$$ Prove that $f(f(x))=x$ has no real roots as well. What I've done is, calculating $f(f(x))$: ...
11
votes
8answers
3k views

How to become proficient in Calculus?

It has been a while since I wanted to ask this question, but couldn't find a right forum. My question might come across as trivial, but its important to me to find an answer to that. Let me give you ...
11
votes
6answers
389 views

Elementary proof that $\pi < \sqrt{5} + 1$

I wanted to show that $$ \frac{\pi}{4\phi} < \frac{1}{2} $$ Where $\phi$ is the golden ratio. I have confirmed the results numerically, and by simple algebra the inequality simplifies down to $$ ...
11
votes
2answers
262 views

A polynomial determined by two values

From a St. Petersburg school olympiad, 11th grade. Prove or disprove: a non constant polynomial $P$ with non-negative integer coefficients is uniquely determined by its values $P(2)$ and $P(P(2))$.
11
votes
4answers
550 views

Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction

How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
11
votes
4answers
501 views

If $2^x=3^y=6^{-z}$ then prove that:$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$

If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0 $ then prove that:$$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$ I have tried starting with taking logartithms, but that gives just some more equations. Any ...
11
votes
7answers
709 views

Derivation of the general forms of partial fractions

I'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take (1) If the denominator of the fraction has linear factors: $${5 \over {(x - 2)(x + 3)}} \equiv {A ...
11
votes
5answers
4k views

How can I write an equation that matches any sequence?

One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is: x 0 1 2 y 1 5 9 If ...
11
votes
3answers
184 views

How do we solve $a \le b^{r}-r$ for $r$?

Given two values $a$ and $b$, how should one go about solving the following inequality for $r$: $$a \le b^r -r .$$ Applying $\log_b$ on both sides of the inequality doesn't help me much since that ...
11
votes
1answer
520 views

Interesting negative decimal number notation

I was studying logarithms, and had to solve the problem: If $\log 8 = 0.90$, find $\log 0.125$. I found out the answer to be $-0.90$. That was easy. But my text book has given the answer as: ...
11
votes
2answers
2k views

Relearning from the basics to Calculus and beyond.

Assume someone has very limited knowledge of math. (low level high school, 5-6 years ago) How would they learn from the basics of algebra, geometry and trigonometry to a solid foundation for calculus ...
11
votes
5answers
1k views

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?
11
votes
1answer
2k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
11
votes
4answers
677 views

Proof that $n^3 + 3n^2 + 2n$ is a multiple of $3$.

I'm struggling with this problem: For any natural number $n$, prove that $n^3 + 3n^2 + 2n$ is a multiple of $3$. That $n^3 + 3n^2 + 2n$ is a multiple of $3$ means that: $n^3 + 3n^2 + 2n = 3 ...
11
votes
2answers
7k 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 ...
11
votes
4answers
532 views

Prove maximum value of $(z-xy)(x-yz)(y-zx)$ is $\frac{1}{64}$ given $x,y,z \in (0,1)$

Prove maximum value of $(z-xy)(x-yz)(y-zx)$ is $\frac{1}{64}$ given $x,y,z \in (0,1)$ I can make it $\frac{1}{64}$ by setting $x,y,z = \frac{1}{2}$, but I have no idea how to show that's the maximum. ...
11
votes
5answers
414 views

How to calculate $2^{\sqrt{2}}$ by hand efficiently?

I've been trying to calculate $2^{\sqrt{2}}$ by hand efficiently, but whatever I've tried to do so far fails at some point because I need to use many decimals of $\sqrt{2}$ or $\log(2)$ to get a ...