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    Basic Algebra

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    Nilradicals and Prime Ideals

    Assume R is commutative ring. Prove that the following are equivalent: a) R has exactly 1 prime ideal b) Every element of R is either nilpotent or a unit. c) R/N(R) is a field Here N(R) denotes the nilradical of R. The set of nilpotent elements form an ideal called the nilradical of R

    Nilpotents, Ideals and Nilradicals

    Prove that if R is commutative ring and N=(a1,a2,..am) where each ai is a nilpotent element, then N is a nilpotent ideal, i.e N^n=0 for some positive integer n. Deduce that if the nilradical of R is finitely generated then it is a nilpotent ideal. P.S. the set of nilpotent elements form an ideal which is called nilradical of

    Revenues Using Logarithms

    Equation - R=12log10Days In 10 days revenue = $12 million how long will it take to reach $36m Answer is 1,000 days but I cannot figure out how this happened. I tried 36=12log10days 36/12 = log10days 3 = log10days 3 = days Incorrect

    Coefficients of Quadratic Equations

    Quadratic Equations Write three quadratic equations, with a, b, and c (coefficients of x2, x, and the constant) as: 1. Integers 2. Rational numbers 3. Irrational numbers See attached file for full problem description.

    Simplifying Radicals Variables

    Simplify each radical expression. Assume all variables represent positive real numbers. Here are six problems. √48 √20n^25 4√32m^11 Find the domain of each radical expression. Use interval notation 4√-5x-1 Simplify the expressions involving rational exponents. Assume all variables re

    Tangent Lines and Logarithmic / Exponential Functions

    Tangent Line and Logarithmic/Exponential Functions This project is going to consider the relationship between a point (a, b) on the natural logarithmic and exponential functions and its relationship to the intercept of the tangent line to the respective functions at the point (a, b). Hint: Recall the distance between two

    Solving Equations

    Equation with two solutions. See attached file for full problem description.

    Fermat and Wilson's Theorem

    1. If and are distinct primes prove that for any integer a, Use Fermat's theorem 2. Show that if and are both primes, then 4[ (mod Use Wilson's theorem. 3. Let be an odd prime. Prove that if g is primitive root modulo and (mod is not Use the binomial expansion See attached file for full

    Equations for time and rate problems and their solutions

    52. How can a graph be used to determine how many solutions an equation has? 8. Bobbi picks 1 qt in 20 minutes. Blanche picks 1 qt in 25 minutes. How long would it take them to pick one quart if they work together? 14. The speed of a freight train is 15 mph slower than the speed of a passenger train. The freight train tr

    Determining formula values.

    1) An open-top box is to be constructed from a 4 by 6 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out. a) Find the function V that represents the volume of the box in terms of x. Answer: b) Graph this fu