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# Basic Algebra

### Uncountable Basis

It can be shown that R (the set of all real numbers) is an infinite-dimensional vector space over Q (field of rationals). Is it true that any basis (by basis I mean algebraic basis or Hamel basis) of R over Q has to be uncountable ?

### Z-Modules and Modules Associated with Representations

1)I understand what a standard R-module (ring-module) is, but I have heard talk of modules associated with representations. Could someone please give me some idea of what these are? 2) I am trying to find all modules over Z-the Integers; so far, I have only come up with additive groups. How can I find all others?

### What is the advantage of using exponents rather than radicals?

While the radical symbol is widely used, converting to rational exponents has advantages. Explain an advantage of rational exponents over the radical sign. What is an example of an equation easier to solve as a rational exponent rather than as a radical sign.

### Finding the Discriminant in a Quadratic Equation

When using the quadratic formula to solve a quadratic equation (ax2 + bx + c = 0), the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. What I need to do is figure out how to create three unique equations where the discriminant is positive, zero, or negative. For each case, please explain what t

### Exponential and Logarithmic Functions and Examples

1. Give an example of an exponential function. Convert this exponential function to a logarithmic function, then plot the graph of both functions. 2. Given the following values 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, x, and y, form: A linear equation in one variable A linear equation in two variables A quadratic equation A polynom

### Common and Natural Logarithms

For the exponential function e^x and logarithmic function log x, graphically show the effect if x is doubled. My response needs to include the reasons for graphically representing the effect in a particular way. I also need to scan the plotted graph and post it along with my response. I need to correctly calculate the values

### Exponential and Logarithmic Function Graphs

Referring to the graph below (which is attached), identify the graph that represents the corresponding function. I need to justify my answer. y = 2^x y = log2x (where the 2 is lower case below log, not above)then; x Also, I need to plot the graphs of the following functions and show them. f(x)=6^x f(x)=3^x - 2 f(x)

### Creation of Quadratic Equations

When using the quadratic formula to solve a quadratic equation (ax2 + bx + c = 0), the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. Create three unique equations where the discriminant is positive, zero, or negative. For each case, explain what this value means to the graph of y = ax2 + bx + c

### See problem below ...

It is approximately 300 miles from Chicago, Illinois, to St. Louis, Missouri. Allowing for various traffic conditions, a driver can average approximately 60 miles per hour. a)Write a linear function that expresses the distance traveled, d, as a function of time, t. Answer: b)How far have you traveled after 3 hour

### Algebra Basics, Equations, and Mathematical Models

Need comparison. 1. When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imagina

### Rectangular Beam : Finding Length Given Cross-Sectional Area, Logitudinal Compressive Force and Unit Strain

Rectangular beam has cross sectional area of 2" x 4". The beam is subjected to a longitudinal compressive force of 6000 lbs. The unit strain is 1 x 10-6 ft/ft. What is the original length if the beam is compressed 1 x 10-6 ft?

### Sturm-Liouville Problems : Eigenvalues, Eigenfunctions and Square Norm

Please help with the following problems. 1. Find the eigenvalues and eigenfunctions of the boundary- value problem y''+lambay = 0 , y(0) = 0 , y(pi/4) = 0 ( n = pi) 2. Find the eigenvalues and eigenfunctions of the boundary- value problem y''+(lamba+1) y = 0 , y'(0) = 0 , y'(1) = 0 3. Find the square norm

### Quadratic equations

1. When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Questi

### Applications of Logarithmic Equations : Earthquakes and the Richter Scale

The intensity level of an earthquake is based on the Richter scale. Using logarithms, the Richter scale measures an earthquake relative to (as a ratio of) the weakest possible tremor. Using web resources, and other course materials, research how earthquakes are measured. Include the following items: 1. What is the formul

### Ellipses and Parabolas

See the attached file. 1. Find the equation of the parabola with vertex at the origin, that passes through the point (-6,4) and opens upward. X=1/9y^2 Y=-1/9x^2 X=-1/9y^2 Y=1/9x^2 2. Find the equation for the parabola with the given vertex that passes through the given point: vertex: (-5,5) point: (-3,17)

### Applications of Logarithmic Equations : Earthquakes and the Richter Scale

The intentisty level of an earthquake is based on the Richter scale. Using logarithms, the Richter scale measures an earthquake relative to (as a ratio of) the weaket possible tremor. Research how earthquakes are measured. What is the formula for measuring earthquakes? ? Why is a 7.0 earthquake ten times stronger than

### Applications of Logarithms

4) For a fixed rate, a fixed principal amount, and a fixed compounding cycle, the return is an exponential function of time. Using the formula, , let r = 8%, P = 1, and n = 1 and give the coordinates (t,A) for the points where t = 0, 1, 2, 3, 4. a) Show coordinates in this space. Show work in this space. b) Show graph

### Cube root of x^2 = 9. solve for x

Cube root of x^2 = 9. solve for x

### Radical and Rational Exponent

Radical and rational exponent notation are two ways to show the same process. Explain the similarities between radicals and rational exponent notation. Provide at least two other examples of mathematical notation or wording denoting the same process.

### Rational Expressions Applications Word Problem

In a cancer research lab, you are observing the growth of cancer cells in two specimens, A & B. Specimen A presently has 2160 cells and specimen B has 1200 cells. After recording data for several days, you later determine that the number of cells in specimen A, N(A), is growing ny the formula: N(A) = 24t^2 + 588t + 2160 w

### You have recently published a research article on the impact of air friction on falling bodies. In your article, you have noted that the deceleration due to friction is a function of the ratio of the volume of the object to the total surface area of that object.

You have recently published a research article on the impact of air friction on falling bodies. In your article, you have noted that the deceleration due to friction is a function of the ratio of the volume of the object to the total surface area of that object. a. Express this ratio for a rectangular prism of length L, widt

### Radicals and Rational Exponents, Pythagorean Theorem, Identity and Side of a Cube Given Volume

1) Solve the following equations. a) Answer: Show work in this space. b) Answer: Show work in this space. c) Answer: Show work in this space. 2) Is an identity (true for all values of x)? Answer: Explain your answer in this space. 3) For the equation , perf

### Radicals and Rational Exponents

Radical and rational exponent notation are two ways to show the same process. ? Explain the similarities between radicals and rational exponent notation. ? Provide at least two other examples of mathematical notation or wording denoting the same process.

### Algebra : Solving Equations and Inequalities

1) What is the solution for x (7x+1)/11=(10x-12)/6 2) What it the solution for x 2<1-4x greater or = to 8 3) Completely factored form 6x^2+21x-12 4) What it the solution for x X^2+4x-12=0 5) What it the solution for x 6x^2+8x=-1 6) What is the slope of the line between these two points (8,1) (-2,5) 7) Fi

### Radical and Rational Exponent Notation

Radical and rational exponent notation are two ways to show the same process. Explain the similarities between radicals and rational exponent notation. Provide at least two other examples of mathematical notation or wording denoting the same process.

### Product of Two Infinite Exponential Series

Prove the following product explicitly using infinite series expressions: Here inf = infinity ( Sum_{k=0}^inf u^k/k! ) ( Sum_{l=0}^inf v^l/l! ) = Sum_{m=0}^inf (u+v)^m/m! Please see the attached file for the fully formatted problems.

### College Algebra : Application Ratio and Proportion Word Problems

The weight of an object varies inversely as the square of its distance from the center of the earth. If an object 8,000 miles from the center of the earth weighs 90 pounds, find its weight when it is 12,000 miles from the center of the earth. Please see the attached file for the fully formatted problems.

### Logarithmic functions on word problems

This problem set is comprised of 8 questions involving logarithmic functions. Complete description of the problem can be found in the attached pdf file. 1. Question involves log to the base of 10 2. Question involves log to the base of 2 3. Question involves finding Decibel Levels 4. Word problem on Earthquake Intensity

### Quadratic Functions : Solving, Factoring anf Applications

1) Using the quadratic equation x2 - 3x + 2 = 0, perform the following tasks: a) Solve by factoring. b) Solve by completing the square. c) Solve by using the quadratic formula. 2) For the function y = x2 - 6x + 8, perform the following tasks: a) Put the function in the form y = a(x - h)2 + k. b) What is the l

### Quadratic Equations: Discriminants and Imaginary (Complex) Numbers

When using the quadratic formula to solve a quadratic equation ax2 + bx + c = 0, the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. (When the discriminate is negative, then we have the square root of a negative number. This is called an imaginary number, sqrt(-1) = i. ) Explain what the value