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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7
votes
1answer
347 views

Finding $\sum\limits_{n=1}^{9999} \frac{1}{(\sqrt{n+1}+\sqrt{n})(\sqrt[4]{n}+\sqrt[4]{n+1})} $

How we can find $$\sum_{n=1}^{9999} \frac{1}{(\sqrt{n+1}+\sqrt{n})(\sqrt[4]{n}+\sqrt[4]{n+1})} $$
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 ...
2
votes
1answer
136 views

Trig question I don't really understand

$4\cos^2 \left( x + \dfrac{1}{4}\pi \right)$ = 3 My final answer: $ x = \frac{11}{12}\pi+k\pi $ and $x = \frac{7}{12}\pi + k\pi $ In the correction model it is $x = \frac{7}{12}\pi + k\pi $ and ...
1
vote
1answer
170 views

How to find the roots of this polynomial

The polynomial: $8x^4-8x^2+1=\frac{\sqrt{3}}{2}$ I can simplify with $u=x^{2}$ to $8u^2-8u+{\frac{\sqrt{3}}{2}}=0$ Mistake $\left(1-\frac{\sqrt{3}}{2}\right)$ apply the quadratic formula: ...
0
votes
1answer
67 views

Is my answer to this trig question correct

$4\cos^2 \left( x + \dfrac{1}{4}\pi \right)$ = 3 My final answer: $ x = \frac{11}{12}\pi+k\pi $ and $x = \frac{7}{12}\pi + k\pi $ In the correction model it is $x = \frac{7}{12}\pi + k\pi $ and ...
0
votes
3answers
83 views

Roots of fractions

For example: Why is $\sqrt{\dfrac{1}{2}} = \dfrac{1}{2}\sqrt{2}$ ? How do you solve (simple) roots of fractions? I'm having trouble because I need this for my upcoming trig test but I haven't done ...
0
votes
2answers
88 views

Lowest possible price before any discount

I am having difficulty solving the following problem A toy store regularly sells all stocks at a discount price of 20% to 40%. If an additional 25% were deducted from the discount price what would ...
5
votes
1answer
1k views

When does the summation of a quotient equal the quotient of summations?

That is, under what conditions would $$ \sum_{i = 1}^n \frac{a_i}{b_i}= \frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i} $$ be true? What about for infinite summations, i.e. when $n \rightarrow ...
1
vote
1answer
74 views

Inequalities of summations

I am thinking if the following condition is in general true: $\frac{n}{m} \leq \frac{\sum_{i = 1} ^ {n} a_i}{\sum_{i = 1} ^ {m} b_i}$, when $n \leq m$ and $a_i \geq 0$, $b_i \geq 0$ but i cannot find ...
0
votes
1answer
67 views

How old will jack be in $5$ years

The question is Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill , how old will Jack be in 5 years? (Ans=23) Here is how I am solving it could anyone tell ...
2
votes
2answers
974 views

Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$

I would like to verify my proof of the following: Let $A=\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$. I want to show that $-1$ and $1$ are the infimum and supremum respectively. First I will ...
0
votes
2answers
55 views

How much more time will it take to finish the filling the pool?

The question is: An empty pool being filled with water at a constant rate takes 8 hours to fill to \frac{3}{5} of its capacity.How much more time will it take to finish the filling the pool? ...
0
votes
1answer
63 views

Found one root, how do I know to keep searching or not?

I'm to solve the equation $$\ln(9t+45) - \ln(5-t) = \ln(t+3)^2$$ After some work I arrive at this: $$t(t^3-4t^2+25t-35) = 0$$ which clearly shows that $0$ is a root for $t$. This is also clear ...
0
votes
3answers
70 views

Mathematical 'language' (geometry)

What does this question mean: 'Show (translated from my native language) that the equation $ x^2 - 4x + y^2 + 6y = 51 $ is a circle.' I have absolutely no idea how to 'show/prove/etc.' it, other than ...
1
vote
2answers
52 views

An analytic geometry question + algebra

We have a Cartesian coordinate system with the points M (a,b) Q (4,2) and P (x,y) but I don't think you need P to solve this one, only M and Q. M is the middle of a circle with a radius r, and Q is a ...
1
vote
3answers
309 views

Solving $\sqrt{(x-2)^2 + (y-1)^2} = \sqrt 2$

What is the answer of this: $\sqrt{(x-2)^2 + (y-1)^2} = \sqrt 2$
2
votes
2answers
161 views

Finding $\alpha$ for $\sin(4 \alpha + \frac{\pi}{6}) = \sin (2\alpha + \frac{\pi}{5})$

I try to find $\alpha$ for $\sin(4 \alpha + \frac{\pi}{6}) = \sin (2\alpha + \frac{\pi}{5})$. Left side: $$\sin(4 \alpha + \frac{\pi}{6}) =$$ $$= \sin4\alpha \times \cos \frac{\pi}{6} + \cos 4\alpha ...
1
vote
1answer
39 views

Two unrelated equations, $w^2 = -\frac{15}{4} - 2i$ and $z^2 - (3-4i)z + (2-4i) = 0$?

I am to solve the equation $z^2 - (3-4i)z + (2-4i) = 0$, and also have to help me that $w^2 = -\frac{15}{4} - 2i$. I can ofcourse find $w$, but I fail to see how this helps me in solving $z^2 - ...
1
vote
2answers
85 views

Number Problems

Benjamin makes up present boxes for children, each box must contain two balloons, One whistle and one tube of sweets. Balloons are sold in packets of 40, whistles are sold in packets of 6 and tubes of ...
3
votes
4answers
266 views

Equality $2^ \sqrt{2\lg n}$ = $n^ \sqrt{\left(\frac{2}{\lg n}\right)}$

Can anybody help me please to prove this equation? $2^ \sqrt{2\lg n}$ = $n^ \sqrt{\left(\frac{2}{\lg n}\right)}$
0
votes
2answers
864 views

how many items can i fit in my box

I have a cuboid with a width of 500mm, a height of 500mm and a length of 1000mm. I need to know how many 38mm balls i will need to fill the cuboid. Any help would be greatly appreciated. Thank you. ...
1
vote
3answers
38 views

What is the right approach to use for factoring a rational inequality

Here is an example: $\frac{x^2 - 3x + 2}{x + 1} -5 > 0$ My approach would be to factor, find the undefined areas and the zeros, and then pick some points in the intervals left to see what I find. ...
3
votes
3answers
293 views

How come there is only one solution to $\sqrt{110-n} = n$

Given: $\sqrt{110-n} = n$ It follows that: $110 - n = n^2$ $n^2 + n - 110 = 0$ $(n + 11)(n - 10)=0$ If $n = 10$, $\sqrt{110 - 10} = \sqrt{100} = 10$, it checks out. If $n = -11$, $\sqrt{110 - ...
1
vote
4answers
169 views

Let $f (x) = 2^x$. Show that…

Let $f(x) = 2^x$. Show that $\dfrac{f(x+h) - f(x)}{h} = \dfrac{2^x(2^h-1)}{h}$. First day of my precalc class in college, and I have no idea where to start to solve this one. Can anyone point me in ...
2
votes
3answers
420 views

Solving a system of equations with natural numbers?

first of all I am sorry if the level of this question is nowhere near the usual level of questions on this site because my math knowledge is still very basic. I hope you won't mind. I found this ...
1
vote
1answer
385 views

How to get the sum/difference of functions

Can someone please explain how to solve this question: $f(x)=−3+2\cos(x)$ $g(x)=\cos(x−\pi/4)−2$ Sum of functions: $s(x)=f(x)+g(x) $ Difference of functions: $v(x)=f(x)−g(x)$ Get the sum and ...
1
vote
0answers
63 views

How to get the sum of functions

I'm having a tough time with this question: $f(x) = -3 + 2\cos(x) $ $g(x) = \cos(x- \frac{\pi}4) -2$ Sum of functions: $s(x) = f(x) + g(x)$ Difference of functions: $v(x) = f(x)-g(x)$ Get the sum ...
7
votes
1answer
346 views

infinity sum of numbers [closed]

If we have a series of numbers $$1^5 + 2^5 + 3^5 + \cdots + (10^n)^5.$$ Final sum of the series is approximately equal $16666\ldots$ . If there is more and more numbers in the series is the ...
0
votes
0answers
83 views

Eccentricity and Length of Semi Axes of a conic

If a conic $ax^2+by^2+2hxy+2gx+2fy+c=0$ and say: How to find the eccentricity and the semi-axes of this conic. I do understand that if its a hyperbola only one of the semi axes will be real. Soham
1
vote
1answer
807 views

How many different 20-person committees can be formed…

Please help me how to solve this question It's just confusing my mind. Q)How many different 20 persons committees can be formed each containing at least 2 professors and at least 3 Associate ...
1
vote
2answers
175 views

Three inequalities with sums of fractions over two positive integers

In a proof, I arrive at three inequalities for all $p,q \geqslant 0$: \begin{align} \frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 + \frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr ...
9
votes
3answers
1k views

Simplify $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$.

Denest $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$. I have tried completing square by several method but all failed. Can anyone help me please? Thank you. p.s. I'm a poor question-tagger.
0
votes
5answers
1k views

How are different functions bounded if they have an asymptote?

Does a horizontal line count as bounded? Also, how is $y=2^x$ bounded?
4
votes
1answer
292 views

Upper and lower limits of $f(x)=\cos(x^2)-\cos((x+1)^2)$ as $x\to\infty$.

I believe that the upper limit is +2 and the lower limit is -2. We have the trigonometric identity $\cos(x^2)-\cos((x+1)^2)=2\sin(x^2+x+1/2)\sin(x+1/2)$. We then make the substitution $x=m2\sqrt{\pi}$ ...
1
vote
2answers
49 views

separable equations.. algebra?

So, I am failing to understand some potentially simple algebra here. I have a separable equation: $ \frac{dy}{dx} = \frac{e^{-x} - e^x}{3+4y} $ and after the easy integration $=>$ $3y + 2y^2 = ...
2
votes
2answers
187 views

Finding highest possible value of function

I am to determine the highest possible value of $p(t)$ when $$p(t) = 7t - 11 - 2t^2$$ I try completing the square: $$= -2\left(t^2 -\frac{7}{2}t + \frac{11}{2}\right)$$ $$= ...
2
votes
1answer
970 views

How to calculate minimum distance between two arbitrary ellipses in 2D?

Arbitrary ellipses means that they can be scaled, translated and rotated in any way in 2D. Do you know some high-school method (might be slightly more advanced than that) to find the minimum distance? ...
0
votes
0answers
60 views

Two bivariate rational functions

I have $2$ bivariate rational equations: $\dfrac{t(b_2^2-a_2^2-t^2b_2^2)+t^3a_2^2}{1+t^2}+\dfrac{ta_1a_2-ts^2a_1a_2-b_1b_2s+b_1b_2st^2}{1+s^2}=0$ ...
0
votes
1answer
314 views

Mathematics Word Problem (cannot work it out for the life of me)

First part of the question A bank was robbed this morning. A lone robber carried the loot away in a big leather bag. The manager of the bank said that the money stolen consisted of small bills: \$5, ...
0
votes
3answers
118 views

Calculating Operating Costs to give a fixed profit margin

We are importing sales data from a company we just purchased. We want to calculate the logistics value so that the profit margin is 13%. We have the following rules in our system: ...
5
votes
3answers
1k views

Factoring the expression $x^6 + 64$

Alright, so apparently I've factored this out wrong... $x^6 + 64 =$ $x^6 + 2^6$ Then I continued, using $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ to get ... $$(x^2)^3 + 64 = (x^2)^3 + 4^3 = (x^2 + ...
1
vote
4answers
599 views

Work and Time calculation

A is thrice as good as a workman as $B$ and therefore is able to finish a job in $60$ days less than $B$. How much time will they take to finish the same job if they work together? My attempt: Let's ...
1
vote
1answer
207 views

The Way to solve an equation involving logarithms

How can I solve the following equation: $$2^{log_3 x}+x^{log_3 2}=4$$ I don't want the final answer, I want to know how I can solve these kind of equations.
1
vote
1answer
368 views

Find the value of $A$ and $B$ such that $P$ is a rational number

Well, I got stuck in that question: Consider the integral expression in $x$: $P = x^3+x^2+ax+1$ where $a$ is a rational number. At $a=A$ the value of $P$ is a rational number for any $x$ which ...
19
votes
6answers
2k views

How to solve $x^3=-1$?

How to solve $x^3=-1$? I got following: $x^3=-1$ $x=(-1)^{\frac{1}{3}}$ $x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
3
votes
2answers
181 views

The value of $\bigl\lfloor x\bigr\rfloor+\bigl\lfloor x^2\bigr\rfloor+\bigl\lfloor x^3\bigr\rfloor+\bigl\lfloor x^4\bigr\rfloor$

There was a multiple choices saying: Find the value of $\bigl\lfloor x\bigr\rfloor+\bigl\lfloor x^2\bigr\rfloor+\bigl\lfloor x^3\bigr\rfloor+\bigl\lfloor x^4\bigr\rfloor$ knowing that ...
7
votes
4answers
321 views

How would I solve $(x-3)(x-2)(x-1)\gt0$?

This problem is on my Calculus Readiness Test and I was having a lot of trouble with it. The problem is $$(x-3)(x-2)(x-1)\gt0$$ I know how to solve $(x-3)\gt0$ but I have never seen these type of ...
-4
votes
4answers
138 views

$\frac{2x + 10}{2} - x = 5$: How do I show that all kinds numbers satisfy this equation?

I want to show that this magic equation is true for all kinds numbers (rational, irrational, imaginary, natural, etc..) which is true as far as I've ...
2
votes
5answers
642 views

Converting an IF condition to a mathematical equation

I am trying to study about converting algorithms into mathematical equations. For this I just started with a simple random example : ...
2
votes
3answers
715 views

GRE Algebra question

A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12. (a) What is the fixed total cost ...