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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1answer
35 views

equation $f(x,y)=0$ can be solved for $y$ in terms of $x$

The equation $f(x,y)=0$ can be solved for $y$ in terms of $x$. What does this expression mean? Sorry for such type of question because English is not my native.
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1answer
77 views

integer solutions for $\left( a+\sqrt{b} \right) ^ n=p+q\sqrt{b}$?

Given whole numbers $a$, $b$, and $n$, where $\sqrt{b} \not\in \mathbb{N}$, there should be a unique solution where $p$ and $q$ are also whole numbers. Is there any way of expressing $p$ and $q$ in ...
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2answers
48 views

Proving trig identity using De Moivre's Theorem

Question: Prove $$\cos(3x) = \cos^3(x) - 3\cos(x)\sin^2(x) $$ by using De'Moivres Theorem So far (learning complex numbers at the moment) that De Moivre's theorem states that if $z$ $=$ ...
4
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3answers
72 views

Showing $\sum_{k = 2}^n \binom{k}{2} \binom{n}{k} = \binom{n}{2} 2^n$ without induction.

How do I prove the identity$$\sum_{k = 2}^n \binom{k}{2} \binom{n}{k} = \binom{n}{2} 2^{n-2}$$combinatorially, i.e. counting the cardinality of the same set in two different ways? I know how to do it ...
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2answers
64 views

At least $P(m, n - 1) = {{m!}\over{(m - n+1)!}}$ surjective functions from $[m]$ to $[n]$?

How do I see that there are at least$$P(m, n - 1) = {{m!}\over{(m - n+1)!}}$$surjective functions from $[m]$ to $[n]$?
4
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1answer
93 views

How many integers between $1$ and $2016$ are divisible by a nontrivial cube $p^3$, $p > 1$? [closed]

How many integers between $1$ and $2016$ are divisible by a nontrivial cube $p^3$, $p > 1$?
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1answer
20 views

The length of a rectangle is $4$ m longer than its width. If the area is $8$ m$^2$, find the rectangle's dimensions. [closed]

The length of the rectangle is $4$ m longer than its width. If the area is $8~\text{m}^2$, find the rectangle's dimensions. Round to the nearest $10$th of a meter. I have absolutely no clue how to ...
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0answers
33 views

Difference of prime products

Using the first seven primes $(2,3,5,7,11,13,17)$, the largest difference that can be created from products of these primes is $5\cdot 7\cdot 11\cdot 13\cdot 17-2\cdot 3=85079$. What is the 2nd ...
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1answer
25 views

What is the equation of the bottom half of the parabola $x + (y - 2)^2 = 0$?

A parabola has the equation: $$x + (y - 2)^2 = 0$$ I can't find the $y$ without getting the equation into some weird recursion.
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2answers
53 views

Calculus: Integration problem.

Does there exist a function $f(x) \neq \dfrac{1}{x}$ such that $\int_1^{\infty} f(x) \mathrm d{x} = \log t$ for all $t>0$ ? Note: the exception $f(x) = \dfrac{1}{x}$ was treated as the limit of ...
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0answers
23 views

$x\in\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$, s.t. $x^2=p+q\sqrt{30}$

Does there exists $x\in\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ of the form $$ x=a+b\sqrt{2}+c\sqrt{3}+d\sqrt{5}+e\sqrt{6}+f\sqrt{10}+g\sqrt{15}+h\sqrt{30},\tag{1} $$ where ...
3
votes
2answers
105 views

$3^x-2^y=17$ where $x$ and $y$ are both positive integers

First, I was wondering what would the solution(s) be to the equation $3^x-2^y=17$ such that $x$ and $y$ are both positive integers. I couldn't find any small possibilities. Is there a proof that such ...
0
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2answers
39 views

Finding an angle between two vectors

I am trying to answer part $d)$ by using my answer to part $c)$. From what I can see, the only possible way to do this is to find the lenght of $AB$ and $OB$, and, using the angle in part $c)$, apply ...
1
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1answer
44 views

Prove that if $f : F^4 → F^2$ is linear and $\ker f =\{ (x_1, x_2, x_3, x_4)^T: x_1 = 3x_2,\ x_3 = 7x_4\}$ then $f$ is surjective

Prove that if $f : F^4 → F^2$ is a linear map such that $$\ker (f)= \big\{ (x_1, x_2, x_3, x_4)^T\ :\ x_1 = 3x_2,\ x_3 = 7x_4\big\}$$ then $f$ is surjective. I know that all $x_1,x_2,x_3,x_4$ ...
2
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2answers
31 views

Solving a radical equation with trinomials on both sides

$$8\sqrt{a^2-4a-16}=3a^2-12a-64$$ I do know the standard procedure—square both sides, isolate square root, square again, check solutions to make sure they are real, etc. However, for a problem such ...
0
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1answer
30 views

Solve equation for t

$$s = 2 \ln|\tan(t) + \sec(t)|$$ I tried to solve it and got a quadratic equation which turned out to equal $arcsin(\dfrac{-2 \pm e^s}{2(1+e^s)})$ This doesn't seem right. Any thoughts?
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0answers
18 views

Prove that the value of $q$ cannot lie between $p$ and $\frac{p(n+1)^2}{(n-1)^2}$.

QUESTION: If $p$ be the first of $n$ arithmetic means between two numbers, and $q$ the first of $n$ harmonic means between the same two numbers, prove that the value of $q$ cannot lie between ...
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8answers
218 views

Prove that $ { a }^{ 2 }+2ab+{ b }^{ 2}\ge 0$ without using $(a+b) ^{ 2 }$

Prove that $${ a }^{ 2 }+2ab+{ b }^{ 2 }\ge 0,\quad\text{for all }a,b\in \mathbb R $$ without using $(a+b)^{2}$. My teacher challenged me to solve this question from any where. He said you ...
5
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6answers
90 views

How is $2^{\log_4 n}= n^{\log _42}$?

I saw in a notebook the following: $2^{\log_4 n}= n^{\log _42}(=\sqrt n)$, but I never saw this before and I can't find it in any log rules, is it right? and if so how did they do it? BTW, if we take ...
2
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1answer
48 views

Find $N$ so that the sequence is the product of three consecutive numbers

Find the smallest natural number $N$ such that $13 \cdot 17 \cdot N$ is the product of three consecutive natural numbers. $x(x+1)(x+2) = 13 \cdot 17 \cdot N$. So let $x=N$, then, $N+1 = 13$ and ...
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2answers
34 views

How many cards need to be picked at least?

You have $50$ cards and you have the numbers from $1$ to $50$ written on them, and you randomly pick cards. How many cards do you need to pick out so you can ensure that at least $3$ cards with ...
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2answers
56 views

How many three digit numbers exist such that the third digit is the geo mean

How many three digit numbers exist such that one of the digits is the geometric mean of the other two? A 12, B 18, C 24, D other So, $N = 100a + 10b + c$ let $c =\sqrt{ab}$. $ab$ must be a ...
7
votes
2answers
634 views

Prove that greatest common divisor of two numbers multiplied with itself divides the product of those numbers

$a, b, c \in \mathbb{N} $ if $c$ is the greatest common divisor of $a$ and $b$, $c^2$ divides $a\cdot b$. $c = \gcd(a, b) \implies c^2|ab $ How would I prove this? I understand why this sentence is ...
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2answers
54 views

Largest range of a list of five integers in which the mean, median and mode are consecutive integers

A student notices that in a list of five integers, the mean, median and mode were consecutive integers in ascending order. What is the largest range possible for these five integers and why? Please ...
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5answers
110 views

How can I compare the numbers $2^{39}$, $5^{19}$ and $52^7$?

I have to compare the numbers $2^{39}$, $5^{19}$ and $52^7$. I don't know how to do that because their exponents don't have anything in common.
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0answers
54 views

How to know if there is a formula to find an answer?

I'm working on a problem and I've found all the values to make it work and I'm trying to figure out if there's a formula in there I can use. Here is what I have written out so far. ...
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1answer
30 views

How to represent a $5$ digit number that has $62$ choices per digit?

If you have a $5$ digit number that can be 0-9A-Za-z how would you represent that? total_number_of_records = 5 digits * (10 + 26 + 26) ^ 5 I want to find out ...
2
votes
1answer
21 views

If A is the range of $f(x) = ^{7-x}C_x$ then the no. of reflexive relation from A to A is…

Problem : If A is the range of $f(x) = ^{7-x}C_x$ then the no. of reflexive relation from A to A is (a) $2^6$ (b) $2^{12}$ (c) $2^{16}$ (d)$2^{20}$ My approach : $f(x) = ^{7-x}C_x = ...
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4answers
50 views

Prove if $x > 100$ then $\frac{100}{3-2x} > -1$

If $x > 100$ then $\frac{100}{3-2x} > -1$ Pf: Assume $x>100$ $-2x > -20$ $-2x + 3 > -200$ $\frac{1}{3-2x} > \frac{1}{-200}$ This is what I have done so far, however I'm not ...
2
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2answers
28 views

Inverses of piecewise functions.

For an example, let $f: \mathbb{R}\rightarrow \mathbb{R}, $be defined by$ f(x) = 2x $ when x is rational and $f(x) = 3x$ when x is irrational. Can it simply be concluded that the inverse is ...
0
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4answers
44 views

How to prove b is positive? [closed]

If A * B is positive and I know that A is positive, how do I prove that B is positive? I have to prove it with just basic laws of numbers, e.g. associative property and stuff like that.
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2answers
40 views

Find all $z\in\mathbb{C}$ such that $e^z = 1$.

We write $z=a+ib$. Now, $$1 = e^z = e^{a+ib} = e^a e^{ib} = e^a(\cos b + i\sin b) = e^a\cos b + ie^a\sin b$$ We have $$1 = e^a \cos b \\ 0 = e^a\sin b$$ Now, I don't understand why it has to be ...
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2answers
15 views

Find $a$ and $b$ such that $a-b=c$ and the geometric mean of $a$ and $b$ is $m$.

Given $m,c\in\mathbb{R}^+$, how can I find numbers $a$ and $b$ such that $a-b=c$ and $m = e^\frac{\ln(a)+\ln(b)}{2}$ (i.e., $m$ is the geometric mean of $a$ and $b$). I understand that this doesn't ...
0
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3answers
71 views

Does differentiation of $f(x)=\log(x)$ yield two different results?

The two different results are :$\frac{1}{x}$ and $\frac{-1}{x}$. I read in my book that: $$\frac{d(\log x)}{dx}=\frac{1}{x}$$ where $x>0$ And: $$\frac{d(\log(-x)}{dx}=\frac{1}{x}$$ where ...
3
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2answers
40 views

How much did he spend the last $5$ day $?$

A man has a habit of spending an amount equal to the date on that day . For example Rs.$18$ on the $18$ th of the month and so on. Now on a fine day he is asked how much he had spent the last ...
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1answer
23 views

Limits w/ Greatest Integer and Abs. Value Function

Find the $$\lim _{x\to 2^+}\ {\lfloor x \rfloor - 1\over\lfloor x \rfloor - |x|}$$
3
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1answer
79 views

(Non-continuous) solutions to $f(f(x))=kx$ and $f(x^2)=xf(x)$

Given a fixed non-zero constant $k\in\mathbb{R}$, find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $$f(f(x))=kx\quad\text{and}\quad f\left(x^2\right)=xf(x).$$ If $f$ is continuous, ...
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3answers
33 views

Simplify $\sqrt[3]{36}*\sqrt[6]{\frac{4}{3}}*\sqrt{27}$ writing each factor in index notation

So I rearranged it in index form: $$36^{\frac{1}{3}}*27^{\frac{1}{2}}*({\frac{4}{3}})^{\frac{1}{6}}$$ After this i changed 36 into 4*9 and then 9 into $$3^2$$ So 36 became: ...
2
votes
1answer
55 views

How to calculate $4 \over {{x^4} + {y^4} + {z^4}}$ from $x + y + z = 1$ and other conditions more?

How to calculate $$4 \over {{x^4} + {y^4} + {z^4}}$$ from $$ x + y + z = 1, $$ $$ x^2 + y^2 + z^2 = 9, $$ $$ x^3 + y^3 + z^3 = 1. $$ Alternative answers: A) $1 \over {33}$, B) $2 \over {33}$, C) $4 ...
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1answer
27 views

Power series confusion when multiplying fractions.

I am stuck on the following question. check that the following sum from 0 to infinity converges using power series. sum of $$ 1/((n+(1/2))^2)$$ the next line of work is : $$4/((2n+1)^2)$$ I have ...
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3answers
53 views

Prove that: $2^n < n!$ Using Induction

I'm told to show that $2^n < n!$ using induction This is my attempt at it: BC: $n=4, 2^4 = 16 < 4!$ IH: n = k, $2^k < k!$ IS: try n = k+1 I'm told to only work from one side, so I try ...
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1answer
50 views

What is the largest value of $n$ where $\lg(n) \le 1,000,000$

What is the largest value of $n$ where $\lg(n) \le 1,000,000$ is the question that has been posed in a book I am currently working through and the answer is $2^{10^6}$. However I'm not sure how to get ...
0
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0answers
20 views

Express slope as a function of x

I'm stuck on this problem: Express the slope of the line joining $(1, 0)$ & any point on the graph of the semicircle below the x-axis, centered at the origin with radius one, as a function of ...
1
vote
2answers
24 views

Finding roots of cubic (trig)

The question is By putting $x$ $=$ $\frac 23 cos (\theta)$ Find the exact roots of the equation in terms of $\pi$ $$ 27x^3 - 9x = 1 $$ What I have attempted: $$ ...
2
votes
1answer
39 views

Prove: $ 1\times3 +2\times4 + \cdots + n(n+2) = \frac{1}{6} \times n(n+1)(2n+7)$ using Induction

I'm told to prove this by Mathematical Induction: $ 1\times3 +2\times4 + \cdots + n(n+2) = \frac{1}{6} \times n(n+1)(2n+7)$ This is what I have so far: BC: Try $n=1$: $ 1\times3 +2\times4 + \cdots ...
0
votes
2answers
37 views

Absolute Value Rational Inequalities

Ok so I have the following two inequalities: \begin{equation} \left| \frac{x+6}{x-2}\right| \leq 4 \end{equation} and \begin{equation} \frac{x^2-1}{\left| x+2\right|} \leq 3(1-x) ...
0
votes
2answers
50 views

If $m = 6x + 5$, what equation is equivalent to $(6x + 5)^2 - 10=-18x - 15$ in terms of $m$?

Hey guys at Mathematics Stack Exchange, I have a question for you guys. This question comes from the Khan Academy practice task entitled 'Solve quadratic equations by using structure', so the credit ...
1
vote
2answers
47 views

Equation with radicals and reciprocals

Find all $x\in\mathbb{R}$ satisfying $$x=\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}.$$ Multiply both sides by $x^{1/2}$ to get $$x^{3/2} = \sqrt{x^2-1} + \sqrt{x-1}.$$Making the substitution $a = ...
1
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1answer
40 views

Find integers $x$ and $y$ such that $\frac{27^{x+y}}{9^{xy}}=27$ and $\frac{4^{2xy}}{8^{x+y}}=512$ .

Find all the integers $x$ and $y$ such that : $$\frac{27^{x+y}}{9^{xy}}=27$$ and :$$\frac{4^{2xy}}{8^{x+y}}=512$$ I'm in Algebra two and I feel like there are certain types of math I haven't ...
0
votes
1answer
33 views

Expanding an expression in a certain field

If $\mathbb F_2$ is a field of characteristic $2$, then we have $x+x=y+y=z+z=0$ for all $x,y,z \in \mathbb F_2$. When I expand $(x+y)(y+z)(z+x)$, I get \begin{align} (x+y)(y+z)(z+x) &= ...