Assignments and most of the other material from the exam 1 period will be placed here. Videos from the Khan academy have been selected here. I have tried to find alternatives.

Assignment for session 1: Watch the NJIT video:

While you are doing so, answer the following questions.

1. When evaluating an exponential expression a^b, where b is a positive integer, come up with a rule for the sign of the expression. Apply your rule to (-2)^3 and -3^2, without evaluating them.
2. When evaluating an exponential expression a^b, where b is a negative integer, come up with a rule for the sign of the expression. Apply your rule to (-2)^(-3) and -3^(-2), without evaluating them.

Also watch the other NJIT video:

The instructor in these videos provides verbal descriptions of the rules as she introduces them. Please write the formulas and describe them using English only and NO math symbols either exactly as she has stated them or do so in a way that makes sense to you.

Assignment for session 2: The first 3 videos are really a holdover from session 1. Watch the NJIT video:

When simplifying a rational expression, what are the steps that have to be taken?  Also watch:

How is multiplying and dividing rational expressions similar and different from simplifying a single rational expression? Your last video is:

Finding the LCD is a bit tricky. What is a preliminary step? What characterizes the LCD? Can you come up with an algorithm for finding it? Once the LCD is found, how do you transform each term so that it has the common denominator?

Now lets get in to the actual material that I assigned for session 2.  The Khan academy has a limited section on complex (nested) fractions.

In the NJIT videos, there are 2 methods which coincides with the 2 corresponding webwork assignments. Method I:

What are the steps that are used? Does the following encapsulation make sense to you? Separate numerator and denominator, simplify, combine, simplify

Now we present the 2nd method:

This second method in contrast to the first method looks that entire expression and looks all the fractions appearing in both the numerator and denominator. The LCD for ALL these fractions are found. Once this LCD is found, what are the subsequent steps?

Assignment for session 3: Here is the link for the Khan academy section on rational equations.

Here is the NJIT video:

I added a comment to the video “There is a small error at the very end of this video when the solutions are checked. Besides being good practice to always check solutions, the video could mention that a check this time has a more important function, which is to catch ‘extraneous’ solution.” Find the error and note it. What do I mean by extraneous solutions? Why can they arise when solving rational equations? Finally, this solution technique for rational expressions closely follows one of our methods for solving complex fractions. Which one?  And why? (hint: what is the first step in both methods?)

(Session 4 was a snow day)

Assignment for session 5:  Rational exponents: if you need lots of help, go here for the Khan academy treatment. Otherwise, the following NJIT video may be enough. However, for those of you a bit shaky or rusty with simplifying radical expressions, you may want to cover that material first (go to the first part of the Assignment for session 5.

Once you understand what a fractional exponent means, then you need to practice going back and forth to radicals. In terms of a radical, to what does the denominator of the exponent correspond? To what does the base correspond? How about the numerator of the exponent? If the base is a negative number and the denominator is even, for instance (-16)^(1/4), why is this not a real number? (hint: change to radical, then question becomes is there is real number which when multiplied by itself 4 times gets you -16?)

Assignment for session 6:  Radical expressions. Here are some Khan Academy videos. In the first video about square roots, instead of trying to find a perfect square as a factor, the radicand is broken down into prime factors. Explain the merits of this approach. In the video about the square root of a fraction, the square root of 200 is encountered.

Assignment for session 7: I spent much time searching the internet for good videos on multiplying radical expressions, but was unable to find any that were just at the right level. Here is one that will provide the basic concepts:

Can you put in words how multiplying two single term expressions works (first example)? My suggestion would be something like “Multiply the parts outside the radical. Multiply the radicands (the parts inside the radicals) and simplify the resulting radical. Finally, simplify and rewrite so that you have a number, a variable part and the radical. For the 2nd example, you should think of the procedure as being very similar to multiplying 2 binomials. What is one big difference? (Hint: how many terms do you typically end up with when you multiply 2 binomials? How many terms do you end up with when multiplying 2 “binomial-like” radical expressions similar to this example?) Study the 2 examples involving the quotients. This is one approach. The other approach is presented here. [This Khan academy video also shows you what to do when the denominator is a binomial.] Use the Khan academy approach to do the 2 divisions above. For the 2nd one, should you multiply by sqrt(3) or 4sqrt(3)? Try both and decide which one keeps the numbers the smallest and involves less work.

Assignment for session 8: There is a whole range of levels of radical equations. Here is the most compact one for our needs:

How would you characterize this example? How many radicals were there to start? Once the radical has been eliminated, what kind of an equation do you end up with? How could you rewrite the steps that NJIT provides so that they also apply to the situation that you end up with a linear equation once the radical has been eliminated? When we studied equations when one or more variable expressions appeared in the denominators, I first said that you had to check the solution in the original equation to eliminate “extraneous” solutions. However, I then changed my tune and said that it was enough (assuming you didn’t make any mistake), to check that the candidate solution did not make a zero in any denominator. Radical equations however do require that each candidate solution gets a full check in the original equation. It is not enough to check that the solution does not give a negative radicand when substituted in. The reason is that when we are squaring to get rid of the radical, we could be turning a false statement into a true statement. The simplest example is “1=-1″ is false, but when you square both sides you get 1=1^2=(-1)^2=1” which is now a true statement. If you study the NJIT example carefully, you will see that this is exactly what happened.