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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4
votes
4answers
259 views

An Inequality about power numbers

Which number is larger? $4^{25}$ or $9^{15}$. Why? I know that it used powers of 2 and 3 but how?
8
votes
4answers
239 views

$x<y$ then $x^3<y^3$

I'm looking for a proof to the following theorem: For any $x,y\in R$: $x<y \Rightarrow x^3<y^3$ I'm trying this approach: Let $z = x^3 - y^3 = (x-y)(x^2+xy+y^2) = z_1 z_2$ where $z_1 ...
3
votes
1answer
44 views

For which angles is inequality true

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 6: a) For which angles $\alpha$ is $\sin^4\alpha-\cos^4\alpha > \sin^2\alpha-\cos^2\alpha$ ? b) For which angles ...
0
votes
1answer
32 views

Basic Mixture question

Let's say that I have 50 ounces of 100% sugar water solution. If I add 50 more ounces of pure water to that solution. What percentage of sugar water solution would I have left?
0
votes
3answers
42 views

Trouble with Logarithmic Differentiation

Hey guys I'm trying to find the derivative of this equation using logarithmic differentiation but I'm having some trouble. Wolfram Alpha is giving me different answers and I'm having difficulty ...
3
votes
2answers
137 views

Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
0
votes
2answers
23 views

finding the factors of a function via gcd

Given a function say, $f(x)$, why is it true that if the GCD of the polynomial and its derivative,$f'(x)$, has a common divisor $\geq 1$, then we have a divisor of the polynomial?
0
votes
1answer
17 views

The condition for an algebraic expression to be square.

My textbook on pre-college Algebra mentions the following fact: We have the expression $y^2(h^2-ab)+2y(gh-af)+g^2-ac$. The condition for this expression to be a perfect square is if we have ...
2
votes
3answers
162 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
-1
votes
1answer
48 views

Proportion problem, family milk in dinner [closed]

If a family of $5$ drink $1$ $1/4$ cup of milk with dinner. How much milk in liter does family drink in $7$ dinners?
1
vote
3answers
31 views

Factor Cyclic Polynomial

Factor $(a+b)(b+c)(c+a)+abc$. I know this is a cyclic polynomial, but I don't know how to solve problems like this. What should I do?
1
vote
1answer
70 views

Why does $\log_{4}32 \neq \log _{4}(4 \cdot 8)$

$$\log_{4}32=2.5$$ If $$\log_{a}(b\cdot c) = \log _{a}b + \log_{a}c \,\,\,; (a>0, b>0,c>0, a\neq 1)$$ Then why does $\log_{4}32$ can't be $\log _{4}(4 \cdot 8)= \log_{4}4+\log_{4}8 = ...
-5
votes
1answer
40 views

how much interest [closed]

My granddaughter is the and is getting a settlement of $12000$ for a car accident. This will be put in some you're of fund until she is 18. At first it would go into done you're of interest bearing ...
0
votes
1answer
21 views

Fundamental Understanding of fractions and thier Properties.

I am quite confused about how fractions can be manipulated within other fractions. For exmple, (2 + cot^2x)/(csc^2x) - 1 Why is it possible to rewrite the expression as (1/csc^2x) + (1 + cot^2x / ...
3
votes
3answers
85 views

Factor $3x^2-11xy+6y^2-xz-4yz-2z^2$

This problem is from my Math Challenge II Algebra class, and it's really confusing. How can you factor something like this? Here's the question again: Factor $3x^2-11xy+6y^2-xz-4yz-2z^2$.
0
votes
2answers
37 views

Parametric Equations

$x=3\sin^3t$ $y=3\cos^3t$ How would I even begin to work out this one? I'm supposed to graph it, but I have no clue what how to even start it.
1
vote
5answers
39 views

Verify algebraically that the equation $\frac{\cos(x)}{\sec(x)\sin(x)}=\csc(x)-\sin(x)$ is an identity

I am stuck when I get to this point $\frac{\cos^2(x)}{\sin(x)}$. Am I on the right track? Verify algebraically that the equation is an identity: ...
3
votes
2answers
49 views

The right procedure on difficult related rates problems

I'm pretty sure the sample problems my teacher gives to us violate some article of the Geneva convention. I'm in talks with my embassy about that, but in the mean time maybe you guys could look over ...
0
votes
2answers
35 views

maximum of a cosine equation [closed]

Let the function $h$ be defined by $h(x) = 2 \cos(10x) + 12$. The maximum value of $h$ is attained at which of the following values of $x$? A. $\pi \over 5$   B. $\pi \over 10$   ...
-2
votes
1answer
20 views

Equation Conversion: Polar to Rectangular

Convert the polar equation to rectangular form (rectangular equation) $$r=\frac{9}{1-3\cos(\theta)}$$ I know that $r^2= x^2+y^2, x= r\cos(\theta)$ and $y= r\sin(\theta)$ and $\tan(\theta)= ...
-2
votes
1answer
75 views

Simplify $\frac{1-\sqrt{x+1}}{1+\sqrt{x+1}}$.

Simplify $\dfrac{1-\sqrt{x+1}}{1+\sqrt{x+1}}$. I did times {1− (√x +1)} under and above = {1− (√x +1)} {1− (√x +1)} / 1 . I can not get it smaller. This is wrong, please help.
3
votes
1answer
112 views

Remainder of $\frac{x^{60}+x^{48}+x^{36}+x^{24}+x^{12}+1}{x^{5}+x^{4}+x^{3}+x^{2}+x+1}$

I am trying to find the remainder of the polynomial division $$\frac{x^{60}+x^{48}+x^{36}+x^{24}+x^{12}+1}{x^{5}+x^{4}+x^{3}+x^{2}+x+1}$$ I know that the answer is 6, but I am not getting that when I ...
4
votes
2answers
59 views

How to turn arbitrary fractions into arbitrary egyptian fractions?

I am reading Stillwell's Numbers and Geometry. There is an exercise about Egyptian fractions which is the following: I've tried to do it in the following way - Expressing an arbitrary fraction ...
1
vote
3answers
29 views

Related Rates Ladder Problem with Angles

The problem is as follows: A 13-foot ladder leans against the side of a building, forming an angle θ with the ground. Given that the foot of the ladder is being pulled away from the building at the ...
2
votes
0answers
21 views

Rendering the derivative of composite functions from a graph

I'm on a workbook problem and I want to make sure I'm doing it properly. The problem asks me to find the derivatives of composite functions when given only the graphs of the original functions, here ...
0
votes
1answer
29 views

can you help me solve my menu board dilemma?

If I have a menu board that measures 35 3/8 $\times$ 71 5/8 and I need to cut in 3 equal pieces, what measurements should each piece be?
2
votes
2answers
74 views

Simplify expression $(x\sqrt{y}- y\sqrt{x})/(x\sqrt{y} + y\sqrt{x})$

I'm stuck at the expression: $\displaystyle \frac{x\sqrt{y} -y\sqrt{x}}{x\sqrt{y} + y\sqrt{x}}$. I need to simplify the expression (by making the denominators rational) and this is what I did: ...
1
vote
3answers
46 views

Understanding a question

If c is randomly chosen from the integers 20 to 99, inclusive, what is the probability that $c^3-c$ is divisible by 12? I have not got the question that what is implying by c? How to get ...
0
votes
1answer
40 views

Vector calculation question…

in the formula for the calulation of the angle between 2 vectors $$\cos \theta \overset{\text{def}}= \dfrac{\vec\alpha \cdot \vec\beta}{|\vec \alpha|\cdot |\vec \beta|}$$ is the output angle is ...
1
vote
1answer
23 views

Line not intersecting circle, maximum value of expression involving radius

If line $y+x=2$ do not intersect any member of circles $x^2 + y^2 -ax = 0$ at two distinct points where a is parameter, then maximum value of $|a + 4|$. My try: Since the line does not intersect ...
7
votes
5answers
467 views
+50

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 ...
1
vote
0answers
42 views

Multinomial Theorem

Can I ask for the proof of the Multinomial Theorem? Wikipedia says: For any positive integer ''m'' and any nonnegative integer ''n'', the multinomial formula tells us how a sum with ''m'' terms ...
3
votes
4answers
70 views

Help with composite functions?

Suppose that $u$ and $w$ are defined as follows: $u(x) = x^2 + 9$ $w(x) = \sqrt{x + 8}$ What is: $(u \circ w)(8) = $ $(w \circ u)(8) = $ I missed this in math class. Any help?
1
vote
1answer
53 views

Simplifying equation variables

I have two emissions equations I am working with that are giving me problems. The first problem I was able to solve my self, the second I have not and am not sure if it is possible. Emission ...
2
votes
3answers
75 views

Simplify $x^2 (1-y)=y^2 (1-x)$

It seems that $(x^2)(1-y) = (y^2)(1-x)$ should simplify. Wolfram Alpha says that solutions are $y=x$ and $y=1/(1-x)$. The first seems intuitively true, but I can't see the algebraic path from the ...
2
votes
2answers
45 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
0
votes
1answer
22 views

Guessing pronumeral

$a,b,c$ are three positive whole numbers. Their sum is $117.$ $a$ is a prime number $b$ and $c$ are multiples of $a$ $c \geq b$ What is the greatest possible value of the product $abc$?
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
0
votes
2answers
56 views

give a complete factored form of the polynomial $-6a^5+48a^4+12a$

Give a complete factored form of the polynomial $-6a^5+48a^4+12a$ I have tried solving this equation and I just cant figure it out. Help me, and give me the answer.
0
votes
3answers
43 views

When a fraction is raised to a negative exponent, do you normally transform it to 1 over the fraction, or invert the fraction?

My text shows that $$\left(\frac{3a^2}{4b}\right)^{-3}=\frac{1}{\left(\frac{3a^2}{4b}\right)^{3}}.$$ It also shows that $$\frac{1}{\frac{144}{b}}=\frac{b}{144}.$$ In the first equation, it seems ...
1
vote
0answers
40 views

A tiled floor of a room has dimensions… [closed]

A tiled floor of a room has dimensions $m \times m$ $\small\mbox{m}^2$. Dimensions of the tiles used are $n \times n$ $\small \mbox{m}^2$s. All tiles used are green tiles except diagonal tiles which ...
3
votes
3answers
126 views

Find the maximum value of $xy^2z^3$ given that $x^2 + {y}^2 + {z}^2 = 1$, using AM-GM

I've been struggling with this equation and how to find the maximum value it can take: Maximise $xy^2z^3$ given that $x^2+y^2+z^2 = 1$ The question is from the book Introduction to Inequalities ...
0
votes
0answers
52 views

Solve system of equations for the ratios of the vectors

(Sorry for the bad title, didn't think of a better way to describe the problem). I have a system $\mathbf{A}\in\mathbb{C}$ that forms the problem $\mathbf{Ax}=\mathbf{b}$, for which I want to find an ...
0
votes
1answer
59 views

If $a^{2/3}=b^{2/3}$, then…

If we solve the math in the following way, $$\Big(\frac{a}{b}\Big)^{2/3 } =1$$ Now cube both side we have, $$\Big(\frac{a}{b}\Big)^{2} =1$$ From this conclusion, can we say that, only the ...
0
votes
1answer
20 views

Relationship between constant term and roots

Does anyone know of a relationship between the constant term of a polynomial and the roots of the polynomial? Specifically, if we know the constant term, is it possible for a root which divides the ...
1
vote
3answers
54 views

surjective, but not injective linear transformation

$T$ is a transformation from the set of polynomials on $t$ to the set of polynomials on $t$. So, the input to $T$ should be a polynomial, and the output should be some other polynomial. Two common ...
1
vote
2answers
78 views

Why does $(3^{1/2})(10^{1/2})=30^{1/2}$ but $(3a^2)(10a^2)=30a^4$?

$(3a^2)(10a^2)=30a^4$? In that equation the exponents are added. Why does $(3^{1/2})(10^{1/2})=30^{1/2}$. In that equation the exponents are not added. Why?
2
votes
2answers
42 views

Square root each term (clarification on polynomials?)

So I'm in Algebra 2, and right now we're learning about conic sections (circles/ellipse/etc). I thought some problems in the workbook looked weird, like this one: ...
2
votes
2answers
52 views

Find an expression that represents how a student should save to get her car.

a student decides she wants to save money to buy a used car, which cost 2600. She decides to save 2 cents the first day and double the amount she saves each day thereafter (1st day 2 cents, 2nd day 4 ...
0
votes
1answer
53 views

Is the question right?

Is this question solvable? I think the required race time is not given. So it is quite impossible to evaluate it.