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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13
votes
3answers
490 views

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

Is there any "superlogarithm" or something to solve an equation like this: $$x^x = 10?$$
13
votes
4answers
754 views

Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$ [duplicate]

Yesterday, my uncle asked me this question: Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$. How can we do this? Note that this is not a diophantine ...
13
votes
4answers
581 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 ??
13
votes
3answers
543 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} ...
13
votes
2answers
865 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. ...
13
votes
4answers
802 views

Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
13
votes
5answers
6k 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?
13
votes
6answers
113 views

Why does $n$-time differentiation of product have the same structure as raising sum to $n$th power?

A formula for differentiating a product is well known: $$(ab)'=a'b+ab'.$$ At first sight it doesn't resemble anything interesting. But what if we differentiate twice? We'll get ...
13
votes
4answers
139 views

Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$

Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$ This problem is copied from Math Olympiad Treasures ...
13
votes
4answers
762 views

How find the value of the $x+y$

Question: let $x,y\in \Bbb R $, and such $$\begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases}$$ Find the $x+y$ This problem is from china some BBS My idea: since ...
13
votes
3answers
145 views

$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$

Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost. Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying: $$x_1 + x_2 \dots + x_k = 0$$ $$x_1^2 + x_2^2 ...
12
votes
6answers
825 views

Average of all 6 digit numbers that contain only digits $1,2,3,4,5$

How do I find the average of all $6$ digit numbers which consist of only digits $1,2,3,4$ and $5$? Do I have to list all the possible numbers and then divide the sum by the count? There has to be a ...
12
votes
3answers
557 views

Showing $\sqrt{2}\sqrt{3} $ is greater or less than $ \sqrt{2} + \sqrt{3} $ algebraically

How can we establish algebraically if $\sqrt{2}\sqrt{3}$ is greater than or less than $\sqrt{2} + \sqrt{3}$? I know I can plug the values into any calculator and compare the digits, but that is not ...
12
votes
3answers
1k views

A real solution to a cubic equation

What is the easiest way to find the real solution of the equation $x^3-6x^2+6x-2=0$? I know the solution to be $x=2+2^{2/3}+2^{1/3}$ (Mathematica) but I would like to find it analytically. If ...
12
votes
3answers
1k views

How to solve : $\,8^x=6x$

I am stuck on the following problem which one of my friends gave me: Solve : $\,8^x=6x$. MY ATTEMPTS: We see that $$8^x=6x \implies 2^{3x}=6x.$$ Now I am not sure how to proceed further. ...
12
votes
4answers
528 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 ...
12
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 ...
12
votes
2answers
325 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
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 ...
12
votes
2answers
512 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
5answers
716 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
12
votes
2answers
712 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
2answers
4k 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 ...
12
votes
3answers
192 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 ...
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
223 views

What's $(-1)^{2/3}\; $?

I know that $\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$ But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?
12
votes
3answers
336 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
4answers
711 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 ...
12
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 ...
12
votes
1answer
528 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: ...
12
votes
2answers
436 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
419 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
3answers
365 views

Collatz-ish Olympiad Problem

The following is an Olympiad Competition question, so I expect it to have a pretty solution: For a positive integer $d$, define the sequence: \begin{align} a_0 &= 1\\ a_n &= ...
12
votes
1answer
3k 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 ...
12
votes
1answer
259 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
8k 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 ...
12
votes
2answers
102 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
4answers
551 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. ...
12
votes
1answer
356 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
181 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
291 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
550 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
226 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
324 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
925 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
271 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
6answers
1k views

Proving that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$

How am I suppose to prove that: $$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$$ Do I use the way like how we count $1+2+ \cdots+100$ to estimate? So $1/5050 \lt ...
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 $2\sqrt{3}$ ...