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
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1answer
83 views

What is the domain of $f(x) = \ln(1+x)^{\sin(x)}$?

Wolfram Alpha says that the domain of $$f(x) = \large \ln(1+x)^{\sin(x)}$$ is $x \gt 0$. I can't figure out how we come up with this restrictions. Please help.
7
votes
2answers
130 views

How do I prove that $\exp(\frac{h}{1+h})\leq 1+h$?

I have come across the inequality $$\exp\left(\frac{h}{1+h}\right)\leq 1+h,\quad\forall h>-1,$$ on http://functions.wolfram.com/ElementaryFunctions/Exp/29/. I would like some help proving this. A ...
1
vote
3answers
287 views

Which branches of maths study the ways solving polynomial equations?

I mean , for example , let $$0 = 1 + 2x + 3x^{2} +x^{7} + 19 x^{9 }$$ and we want to solve this equation, what branch of maths doing this? I know that there is no algebraic solution of the problem ...
1
vote
2answers
8k views

Wolfram Alpha: How to define constants in a system of equations?

I'd like to use WA to solve a small system of nonlinear equations, that involve both constants and the variables of interest. How do I "tell" WA which variables are the constants, and which are the ...
3
votes
4answers
154 views

What is the limit of $\frac{e^{6x}-2e^{3x} + 1}{x^2}$, as $x \rightarrow 0$?

I am to calculate $\frac{e^{6x}-2e^{3x} + 1}{x^2}$ when $x$ goes towards $0$. I find that $$\frac{e^{6x}-2e^{3x} + 1}{x^2} = \frac{(e^{3x}-1)^2}{x^2} = \left(\frac{e^{3x}-1}{x}\right)^2$$ ...
1
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2answers
3k views

Represent the following decimal values in 8-bit 2's complement and hexadecimal

-27: 11100101 -128: 10000000 I'm having trouble coming up with the hexadecimal representation. Does it change if my numbers are represented in 2's complement as opposed to just regular binary? ...
0
votes
1answer
150 views

Derivation of Kinematics Equation

$\vec{s}$ is an arbitrary position vector, which can change with time. $\vec{v} = \dfrac{\Delta \vec{s}}{\Delta t}$ is the velocity vector. $\vec{a} = \dfrac{\Delta \vec{v}}{\Delta t}$ is the ...
2
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4answers
424 views

Solve the inequality on the number line?

How would I solve the following inequality. $x^2+10x \gt-24$ How would I solve it and put it in a number line?
1
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3answers
58 views

Solving simultaneous equations

How do you go from $$q_1 = rq_1 + pq_2$$ $$q_2 = rq_2 + pq_3$$ $$q_3 = rq_3 + p $$ to get $$q_1 = \left( \frac{p}{1 - r} \right) ^3 $$ Because of that extra $q_3$, I keep getting lost in what I'm ...
1
vote
1answer
39 views

How do I solve this equation for R/r?

How do I solve this expression for $\frac{R}{r}$? $\frac{1}{(r-R)^3}-\frac{y}{R^2(r-R)}=\frac{1}{r^3}$
0
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1answer
49 views

How to solve this formula?

My math is a bit rusty so please bare with me. I am trying to solve this formula: $$x -(x y) = z, \quad \text{where}\quad y = 0.15 \quad\text{and}\quad z = 10 000.$$ How to get $x$? This is what I ...
0
votes
1answer
134 views

Which of the following statements are true for the sides of a triangle?

Which of the following statements are true? (a) if $a,b$ and $c$ are the sides of a triangle , then $(ab+bc+ca)/(a^2+b^2+c^2)≥(1/2)$ (b) if $a,b$ and $c$ are the sides of a triangle , then ...
0
votes
1answer
65 views

How to solve this Partial Decomposition fraction?[SOLVED]

I was doing my laplace transform revision and I came across this problem. How to decompose this partial fraction ?? $$ \frac{145s}{(s^2+4)(s^2+4s+13)} +\frac{9s+55}{s^2+4s+13}$$ How to write the ...
0
votes
3answers
60 views

Series and Sequences

Is there any formula for the series below: $$ (2^0)n + (2^1)(n-1) + (2^2)(n-2) + (2^3)(n-3) + \cdots +(2^{n-1})(1) $$ If no, please let me know how to solve such series
3
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0answers
308 views

How to solve the following system with parametric equation?

What is the maximum number of solutions for this system? Parameters $a_0, a_1, ..., a_{2013}$ could be any numbers. \begin{cases} y = a_0 + |x - a_1| + |x - a_2| + ... + |x - a_{2013}|\\ x^2 + y^2 = 1 ...
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2answers
56 views

employee scheduling

I have $20$ employees, I am trying to figure out a schedule of how to work them in to a $6$ on $3$ off or $5$ on $3$ off or $4$ on $2$ off schedule (a)on=working (b)off=days off I am trying to ...
1
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3answers
99 views

What step am I missing in this?

I know this expression equals $-4k$, but I cannot figure it out. Can anybody please walk me through this step by step? original expression: $$1-(2(2k+1)-1)$$ so, my first step is: $$1-(4k+2-1)$$ ...
2
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5answers
4k views

Factoring Polynomials with four terms and two variables

I've been working on this for hours and cannot figure it out. When I search, I find factorization techniques that I already know but don't seem to be able to apply here, or that are for polynomials ...
1
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3answers
140 views

How to solve for $x$ in $x^3+8^x-9=0$

How to find real numbers $x$ that are solutions to $$x^3+8^x-9=0$$ Please help me.
3
votes
2answers
2k views

What does lg x mean? is it $\log_2 x$ or $\log_{10} x$ in binary trees

I'm a bit confused, $\log_{10} x = \log x $ right? I believe I've read somewhere that $\log_{2} x = lg x$ but some people say lg = $\log$. So what does lg really stand for? specifically when talking ...
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4answers
158 views

the value of :$ ((a-b)(b-c)(c-a))^2$

If the polynomial : $f(x)=x^3-3x+2$ have the roots :$a,b,c$ How to find the value of :$$ ((a-b)(b-c)(c-a))^2$$
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0answers
68 views

The solution of the following polynomial equation

I want to solve the following equation: $$\frac{bc}{2}t^{4}-bct^{3}+\frac{a}{b^{2}}t^{2}-2\frac{a}{b^{2}}t+\frac{a}{b^{2}}=0$$ Please help me.
4
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0answers
86 views

Polynomial bound

Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that $$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$ Suppose that $P(x)> 0$ for all ...
1
vote
3answers
236 views

Convert the following decimal to both binary and hexadecimal. How do you interpret $-27_{10}$?

I'm having trouble with this problem because I don't know how to interpret the question. I'm given three numbers: $-27_{10}$, $-128_{10}$, and $150_{10}$. How do I interpret these numbers? What ...
1
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3answers
134 views

Finding the integer solutions

Find all integer solutions of $$(a + b^2)(a^2 + b) = (a − b)^3.$$ Obviously $b = 0$ is one. But how to get other solutions?
3
votes
4answers
2k views

Find an equation of the plane

I have come across a difficult problem in my textbook. The problem ask: Find an equation of the plane that passes through the line of intersection of the planes $x-z=1$ and $y+2z=3$ and is ...
2
votes
2answers
108 views

How do we solve the system of equations?18-1-2013

Giải hệ phương trình: (Editor's note: This translates to "Solve a system of equations:") $$ \begin{cases}\sqrt[4]{x}\left(\dfrac{1}{4}+\dfrac{2\sqrt{x}+\sqrt{y}}{x+y}\right)=2 \\[8pt] ...
1
vote
1answer
54 views

Cyclotomic polynomial simplification

I try to work out what $\Phi_{12}(z)$ is: By the fundamental theorem of arithmetic: ...
2
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2answers
79 views

Simplifying a radical

How would I simplify the following radical. $$\large \sqrt[12]{64a^{3}b^{6}}$$ I know that I can do $$\large 64^{\frac{1}{12}}a^{\frac{3}{12}}b^{\frac{6}{12}}$$ But I am unsure how to simplify it ...
2
votes
1answer
74 views

Can this problem be solved or is there something wrong with it?

By selling $45$ lemons for Rs $40$, a man loses $20\%$. How many should he sell for Rs $24$ to gain $20\%$ on the transaction?
2
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1answer
66 views

Linear system with probabilities (algebra)

I have a small problem that delays my project , it seems I am stuck here loads of time. It is probably very easy but I cant see it right now , I am very anxious about this, please take a look: ...
1
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0answers
46 views

find all sequence satisfies $y^n_1 + 2y^n_2 + · · · + 2011y^n_{2011} = a^{n+1}+1$

Determine all sequences $(y_1, y_2, . . . , y_{2011})$ of positive integers such that for every positive integer $n$ there exist an integer a with $y^n_1 + 2y^n_2 + · · · + 2011y^n_{2011} = ...
3
votes
4answers
268 views

Radical questions algebra

Hello everyone how would I simplify the following radicals. $$3 \sqrt{2a^3b^5} \sqrt{32ab^{2}}$$ I got $$3 \sqrt{64a^4b^7}$$ I know $64$ square root is $8$ and $a^4$ square root is $a^2$ My second ...
4
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3answers
99 views

I need to know the formula to compute time to reach a star

I am working on a sci-fy short story and I need to use a somewhat realistic value for the time in transit for going to a star. Assume that energy on the space ship is not a problem. The space ship ...
1
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1answer
59 views

Ideas To Show this statement: $\prod_2^{2n+1}(1-k^{-2})=(n+1)/(2n+1)$

I would like to know an idea on how to show this: $\prod_{k=2}^{2n+1}$ $(1-{1\over k^2})$ = ${n+1\over 2n+1}$ $\forall$ $n\ge$ $2$. I already checked for $2$ and tried it by induction but I didn't ...
3
votes
1answer
125 views

Geometrical representation of $n^n$

How can we represent geometrically $$n^n,\quad n \in \mathbb{N}$$ For example if $n = 1 \longrightarrow$ a line segment of unit length, $n = 2\longrightarrow$ a square of side $2$ unit's...... ...
1
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1answer
160 views

Maximum value of the product ab

Problem 312. What is the maximum value of the product $ab$ if $a$ and $b$ are non-negative numbers such that $a+2b=3$? Solution. It is easier to say when the product of two non-negative numbers $a$ ...
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3answers
52 views

Simplify algebra exponent

I know I have asked a similar question in the past I am stuck on this question. How would I simplify the following: ...
3
votes
4answers
336 views

Exponent questions algebra

How would I solve the following two exponent questions? (1) The first question is $$\left(\frac{x^{-2}+y^{-1}}{xy^2}\right)^{-1}$$ I got $\quad \displaystyle \frac{-xy^{-2}}{x^2+y},\;\;$but this ...
3
votes
3answers
103 views

Solving equation that involves nepper number

I have the following equation. $$\frac{1}{5} = \large e^{\frac{10^{-4}}{x}}$$ I assume that x will be $0.0621$ but I can't remember how does one solves such equation. I recall that it involves using ...
0
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2answers
188 views

Calculating wholesale and shipping costs for book order

Suppose the cover price of a book is $\$$24.95, but bookstores get a 40% discount. Shipping costs $\$$3 for the first copy and 75 cents for each additional copy. What is the total wholesale ...
2
votes
2answers
70 views

An equation in natural numbers

Given $a,b,n\in \mathbb{N}$. What is the easiest route to find a pair of integers $x,y$ such that $(a^2+b^2)^n = x^2 + y^2$?
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4answers
2k views

Can a finite sum of square roots be an integer?

Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer? The square roots need to be irrational.
0
votes
1answer
86 views

Proof of floor of division

I got stuck proving $$\left\lfloor\frac{x/a}b\right\rfloor = \left\lfloor\frac{\lfloor x/a\rfloor}b\right \rfloor$$ This is what I got: Using the division algorithm we can write $x = qa+r$, where ...
2
votes
5answers
105 views

Is it more practice and intuition or rather algorithmic to solve third degree polynomials of this type?

Consider $x^3 - 6x^2 + 11x - 6 = 0$ I can not reasonably factor this intuitively in any short amount of time with my skill level. Is this the only hope to solving such equations by hand? What tools ...
2
votes
3answers
634 views

How do I simplify this fractional expression?

$\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}$ and these numbers are all above the denominator of $h$. Can someone please help me to understand how to simplify this expression?
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4answers
70 views

Question regarding differentiating with $e$

What is $\left[h(x) = \dfrac{3x-2}{e^x} \right]'?$ My textbook tackles this problem in this way: $h'(x) = \dfrac{e^x\cdot3-(3x-2)e^x}{(e^x)^2} = \dfrac{3-(3x-2)}{e^x}$ etc... I however don't ...
1
vote
1answer
444 views

Dividing the lateral area of a cone into two equal parts.

"The height of a cone is h. A plane parallel to the base intersects the axis at a certain point. How far from the vertex must this point be if the plane divides the lateral area into two equal parts? ...
6
votes
0answers
369 views

Series sum formula

Is there any general formula to sum following series: $$S = 1^1 + 2^2 + 3^3 + \dotsb+(n - 1)^{n - 1} + n^n, n \in N$$ I mean for $S = f(n)$, is there a formula to compute $f(n)$? Regards, vishal.
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2answers
134 views

Simplifying the sum of terms that are “polynomial fractions”

I need to simplify: $$\left(\frac{z^2-2}{1+2z}\right)^2 \;+ \;\left(\frac{z^2-2}{1+2z}\right)$$ I get $2z^4-8z^2+8$ in the numerator when I know it should be $((z^2-2) (z^2+2 z-1))$.