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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0
votes
3answers
138 views

Solving equation containing different terms of the form x^x

Is it possible to solve the following equation for $x$ as a function of $y$: $$\sqrt{\frac{x+k}{x}}\,\frac{(x+k)^{x+k}}{x^x}=y$$ in a way that the resulting equation $x=f(y)$ is something I can ...
1
vote
4answers
72 views

If $16^{\sin ^2x}=5$, then what is $2^{\cos^2x}$?

I happened to create this problem and solved it. I used only basic algebra and trigonometry. I thought it was a fun problem, so I wanted to expose the problem to the public. Please provide an exact ...
0
votes
1answer
32 views

Factoring and solving a cubic polynomial

When can we not use synthetic division to solve for a cubic polynomial? For example we can use synthetic division to solve $-t^3 -4t^2 +20t +48$. When I can't use synthetic division what are my other ...
2
votes
1answer
56 views

How many asymptotes does $y=\frac{x^2}{x-2}$ have?

In a question I came upon, the answer insisted that there were three; one was apparently a horizontal asymptote, which I do not agree with. There are only 2 asymptotes, correct? One is $y=x+2$ and the ...
0
votes
1answer
33 views

Solving for single variable proving to be extremely difficult.

I have been at this equation for about two days now, and I can not for the life of me find a way to solve to i. If anyone can please show me a step by step into solving this, it would help me out so ...
-2
votes
1answer
16 views

math work mixed applications [closed]

Martha's aunt bought \$26 for 20 plants. Some were Carnations selling at 4 for \$3.00. The rest were Zinnias priced at 3 for \$5.00. How many plants of each type did she buy?
0
votes
1answer
33 views

Domain of $\left(f(x)\right)^a$ where $a$ is an irrational number.

Why, if $f(x)$ is a real function and $$\left(f(x)\right)^a$$ where $a$ is an irrational number, we put $$f(x)>0$$ for its domain?
3
votes
3answers
41 views

Forming Partial Fractions

Suppose we have: $ \frac{f(x)}{g(x)h(x)} $ and we want to break it down into; $ \frac{I(x)}{g(x)} + \frac{J(x)}{h(x)}$ and that; $deg(f) \leq deg(g)+deg(h)$ , $deg(i) < deg(g)$, $deg(j) ...
2
votes
0answers
28 views

Polynomials and Divisibility Rule.

The question is this - If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^3)+x^2g(x^6)$ is divisible by $x^2+x+1$, then which of the following are true? 1. $f(1)=g(1)$ ...
1
vote
2answers
39 views

Algebra and solving for n

$$162\left(1-\left(\frac{1}{3}\right)^n\right) -162=-0.05$$ Solve for n I've tried myself but am getting 2.something and the answer should be 7.36. I know you need to use logs but not working for me ...
2
votes
1answer
59 views

What did I do wrong trying to find this limit?

In another question, a user asked to find: $$\lim_{x\to 0} \frac{\exp(x^2)-\cos(x)}{\sin(x)^2}$$ I thought I could use pure trigonometric identities to find the limit. Apparently I was mistaken, but I ...
3
votes
1answer
56 views

Find integer $n$ that satisfies $(\lg n)^{2^{100}} <\sqrt{n}$ with $n > 2$

If $(\lg n)^{2^{100}} < {n^{1/2}}$, where $\lg$ is the binary logarithm, then $$(\lg n)^{2^{101}} < n$$ $$2^{101}\lg \lg n < \lg n$$ $$101 < \lg \lg n - \lg \lg \lg n$$ I don't know that ...
0
votes
2answers
60 views

How can I find the derivative of this integral?

A function is defined for a constant $x$ after integrating out with variable $t$ as: $$ F(x) = \int_0^4 \log(1-x^2t^2)\,dt $$ making it as a function of $x$. How can I now find $ F(x=0)$ and $ ...
1
vote
0answers
35 views

Is there a companion to the book 'A Synopsis of Elementary Results in Pure and Applied Mathematics' by George S. Carr?

A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr is as most of you probably know a book that was famously used by the great mathematician Ramanujan. It is said he ...
6
votes
8answers
105 views

How to show that $6^n$ always ends with a $6$ when $n\geq 1$ and $n\in\mathbb{N}$

Is there a proof that for where $n$ is a natural number $$6^n$$ will end with a $6$? I can understand conceptually that $6\cdot 6$ ends with $6$ and then multiplying that by $6$ will still end with ...
-1
votes
2answers
52 views

Finding a single irrational root to a rational function.

The function is: $$y={x^3 + 3x^2 + 6\over x-3}$$ I have to do a sketch and was able to find $y$-int, vertical asymptote, end behavior asymptote. When searching for roots I used rational root ...
1
vote
1answer
30 views

numerical question from practice test

If I may could I ask for help on the following question taken from SHL practice test? Jason is considering purchasing a new machine to make plastic silverware. The machine produces 1,000 pieces of ...
0
votes
3answers
55 views

If $(58)^a=(5.8)^b=10^c$, then what is the relation between $a,b,c$?

How can I solve this, when the indices are not equal. Thanks! Sorry if this is a stupid question, but I'm studying to improve my math.
0
votes
4answers
44 views

Please help solve for the variables A and B

$ 4A - 2B = 10 $ and $ B = 2A - 5 $. Help Solve for the algebraic variables "$A$" and "$B$" using elimination, substitution or graphing from above two equations. Some say it is one equation only, ...
-1
votes
3answers
75 views

Interesting question in internet [duplicate]

Is this even possible to solve? 30 is an even number. I don't think there's Answer for this .
0
votes
0answers
51 views

Find the sum of the coefficients in front of the even degrees of x in the normal form of a polynomial

Find the sum of the coefficients in front of the even degrees of x in the normal form of a polynomial $$(x^6 + x + 1)^{2015} + (x^6 + x - 1)^{2015}$$ I am familiar with the binomial theorem , ...
0
votes
1answer
76 views

Where is my formula false??

I wrote a formula that returned how many numbers in a given row of pascals triangle are divisible by a given prime. This formula was created to answer https://projecteuler.net/problem=148. I was ...
1
vote
0answers
23 views

Writing the possible values of “x” in an equation or making a solution check at the end?

Should I write the possible values of an equation or make a solution check at the end? For example, I have the equation: $ln (x^4-25x^2) - ln(x^2) = 0$ Should I first write that: $x^4 -25x^2 ...
1
vote
1answer
17 views

Growth of modified binomial recurrence

The binomial coefficients $\binom{n}{r}$ satisfies $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$. This means it is a solution of the equation $f(n,r)=f(n-1,r)+f(n-1,r-1)$, with initial conditions ...
3
votes
3answers
58 views

How can i calculate Total no. of digit in $2^{100}\cdot 5^{75}$

How can i calculate Total no. of digit in $2^{100}\cdot 5^{75}$ $\bf{My\; Try::}$ I have used $$\log_{10}(2) = 0.3010$$. Now Total no. of digit in $$x^y = \lfloor \log_{10}x^y\rfloor +1$$ Now ...
1
vote
1answer
38 views

Expanding brackets to power of -1/2

How do you expand this? $$ \left(16-\frac{x^2}{4}\right)^{-\frac12} $$ And generally how would you expand any $ (a+b)^n $ including fractional and negative powers.
0
votes
1answer
31 views

Can limits be broken into parts?

I know that $\frac{f(x)}{f(y)}$doesn't necessarily equal $f(\frac{x}{y})$. But I was wondering for instance when solving for the limit as x approaches 9 of f(x) where $ f(x) = \frac{3 - ...
5
votes
2answers
202 views

Find the sum of all odd numbers between two polynomials

I was asked this question by someone I tutor and was stumped. Find the sum of all odd numbers between $n^2 - 5n + 6$ and $n^2 + n$ for $n \ge 4.$ I wrote a few cases out and tried to find a pattern, ...
3
votes
2answers
23 views

If $n\ge2$, prove that $\frac {n!}{n^n} \le ({\frac 1 2})^k$, where $k$ is the greatest integer $\le \frac n 2$.

Using only precalculus knowledge, if $n\ge2$, prove that $\frac {n!}{n^n} \le ({\frac 1 2})^k$, where $k$ is the greatest integer $\le \frac n 2$. (taken from Apostol's Calculus I, page 46) I don't ...
0
votes
1answer
18 views

Growth of binomial recurrence with different initial conditions

The binomial coefficients $\binom{n}{r}$ satisfies $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$. This means it is a solution of the equation $f(n,r)=f(n-1,r)+f(n-1,r-1)$, with initial conditions ...
-1
votes
2answers
32 views

Simple equation not working?

So I found this equation $$ e^{20}=10^{20*.43429\dots} = 10^{8.6858\dots} = 10^9 * 10*(-.3141\dots) $$ on this webpage and I'm really confused as to why it isn't matching up whenever I try it. The ...
3
votes
3answers
99 views

How to prove that $4^{2n}-1$ is divisible by $3$ or $5$

My task is to prove that $4^{2n}-1$ is divisible by $3$ or $5$, with $n=1,2,3,...$. Any hints? What is the key observation? Thanks :)
3
votes
2answers
30 views

Factor out the following expression $ -5 a^3 b^3 c + 125abc $

Recently I was doing some factoring exercises and encountered the following problem. The idea is to simplify the expression. $$ -5 a^3 b^3 c + 125abc $$ We find the GCD and bring it outside $$ ...
3
votes
2answers
128 views

How many solutions can the $2^a+3^b+4^c+5^d+6^e=22$ equation have, if $a,b,c,d,$ and $e$ are whole numbers? [closed]

How many solutions can the equation $$2^a+3^b+4^c+5^d+6^e=22$$ have, if $a,b,c,d,$ and $e$ are whole numbers? Can you tell me a step by step answer?
1
vote
1answer
25 views

The maximum value of expression $ \sqrt{\sin^2x+ 2a^2} - \sqrt{-1 -\cos^2x+ 2a^2} $

If $a,x\in\Bbb R$, what is the maximum value of the expression $ \sqrt{\sin^2x+ 2a^2} - \sqrt{-1 -\cos^2x+ 2a^2} $? I tried to differentiate but it became messy.
3
votes
3answers
53 views

Rule of 72 doubling time

I need some help understanding this. So as far as I can tell. The rule of 72 is used to determine when prices will double in years. This is done by 72 divided by the rate, or interest. So it would ...
4
votes
1answer
113 views

Clash of arrows [on hold]

As this problem fits a mathematical subject i decided to bring it down here: in this picture , the arrow length in $1/5$ , Side of the polygon is 1 , all the arrows are unleashed simultaneously ...
0
votes
1answer
16 views

Application of Vieta's formula

Could anyone advise me on how to use Vieta's formula to solve the following problem: If $a+b+c= 12, a^2+b^2+c^2=50, a^3+b^3+c^3=168,$ find $a,b,c.$ Thank you.
1
vote
1answer
62 views

Binomial Series. Product series of coefficients

How to solve this question? Please provide hints only.
-3
votes
1answer
43 views

Show that $b^2=9a^2+4ac$ given that the square roots differ by 3 for a quadratic equation $ax^2+bx-c=0$

If the square roots of $ax^2+bx-c=0$ differ by 3, show $b^2=9a^2+4ac$. How do I show this, from the knowledge that I have about alpha and beta, I can't get it right.
0
votes
1answer
19 views

Unique solution of nolinear equation set

$$\left\{ \begin{aligned} f_1(x_1,x_2...x_n)=0 \\ f_2(x_1,x_2...x_n)=0 \\ \vdots \\ f_n(x_1,x_2...x_n)=0 \end{aligned} \right. $$ $f_i\in C^\infty(R^n)$,what is the condition that make the equation ...
0
votes
1answer
18 views

Prove the total surface area of the interior walls of the tank is $A = xy + 64(\frac{1}{x}+\frac{1}{y})$

I have this question that I need to use multivariable calculus to prove: A rectangular tank, which is open at the top has a total volume of $32m^3$. The base has dimensions x and y Show that the ...
0
votes
1answer
26 views

Proving cube volume larger than box volume with a constant surface area.

Apologies for what may likely be a simply question. Given a surface area of $150$, my class was asked to find the maximum volume of a "box" with this surface area. This process was relatively ...
0
votes
0answers
40 views

Modified tanh(x) function

Undoubtedly everyone at this forum is familiar with the function $tanh(x)$: $$f(x) = \frac {e^{x} - e^{-x}}{e^{x} + e^{-x}}$$ It is a popular choice for a sigmoidal function, due to its many ...
1
vote
1answer
40 views

how to calculate the sum of $1^k+2^k+3^k+\dots+n^k$ [duplicate]

how to calculate this?I have some indistinct memory that it can be solved by induction, but I forget how to do it.
1
vote
2answers
24 views

All real values $a$ for a $2$-dimensional vector?

Find all real numbers $a$ for which there exists a $2D$, nonzero vector $v$ such that: $\begin{pmatrix} 2 & 12 \\ 2 & -3 \end{pmatrix} {v} = a {v}$. I substituted $v$ with $\begin{pmatrix} c ...
1
vote
2answers
74 views

If some of your boomerangs don't come back, how many throws will you get? [closed]

Let's say you're practicing throwing boomerangs. You're not an expert, and only 50% of the time does a boomerang return to you. So you stand out in a field with 16 boomerangs and start throwing ...
0
votes
2answers
32 views

Why the $ -x$ in the expression $(-x^2y^3)^2$ becomes positive when simplified in accordance of the rules of exponents

I have this expression $(-x^2y^3)^4$ which results in $x^8y^12$ when simplified in accordance of the third law of the exponents rules which is $(a^m)^n = a^{mn}$. Why the $-x$ becomes positive when ...
2
votes
2answers
23 views

Basic help with factoring

I am having a small problem recalling how to factor with exponents and roots. For example, I understand $\sqrt{16t^2+4t^4}$=$2t\sqrt{4+t^2}$ But I have issues when it is factoring not with a square ...
0
votes
6answers
85 views

Why does $(2^{20}+2^{20}+2^{20}+2^{21})=5\cdot 2^{20}$?

I did this question on artofproblemsolving.com and I do not understand the solution. Why do I have $5 \cdot 2^{20}$? Can anyone explain?