Wednesday 22 February class

Topics:

• Discussion of homework: distinguishing changing to a common denominator from clearing denominators: when do we use each one, and why, and how we do them. Please make sure that you know and understand the difference and can describe what you are doing using these terms! Also we stressed that the division line in a rational expression is also a grouping symbol, so it is very important that you extend it to the right length. Notice how we draw the division lines when we have complex fractions. The grouping is essential!

• Roots and radicals: see below for definitions, vocabulary, and important theorems

Definitions:

square roots:

b is a square root of a

n-th roots (where n is a natural number):

b is an n-th root of a 

Vocabulary:

Considering n-th roots,

n is called the index or degree of the root.

The sign is called the radical sign.

The quantity under the radical sign is called the radicand.

Notation: in all of these, represents  a real number.

represents the principal square root of (the positive square root), if is a positive real number. The other square root will be .

represents the principal n-th root of . Notice that the index is placed inside the angle of the radical sign: it is very important that you write it there and not in front of the radical sign!

There are two cases:

• If n is an even natural number, and is a positive real number, then there are two n-th roots of , and  is the positive n-th root (just as in the case of the square root). The other n-th root will be .

• If n is an odd natural number, then for any real number , there is exactly one n-th root of , and it is represented by .

It is always true that  for any natural number index n.

Theorem: (NOT a definition, despite what your textbook says!)

For any real number x,

if n is an odd natural number

BUT

if n is an even natural number

Illustration: (this is to show why the theorem is true – it’s not a proof, but a “proof”)

Suppose we take x to be  -4

Then by carefully following the order of operations (and recalling that the radical sign is also a grouping symbol, so any operation in the radicand has to be done before we take the root) we can compute the following:

for the odd index 3,

so we get back -4 in the end,

BUT for the even index 2,

, which is , not -4 itself!

BE CAREFUL!

Take a look at Example 5 in section 6.1 to see how the Theorem is used to simplify various radical expressions.

Homework:
• Review the examples and homework problems discussed in class. Make sure that you understand when and why we need to change to a common denominator, and when and why we clear denominators. Also make sure that you understand and can correctly use all of the vocabulary and notation!
• Do the assigned problems from section 6.1 from the Course Outline, only up to #37!
• Do the WeBWork – The assignment has two parts, both due by next Sunday evening, but don’t wait to the last minute!

Reminder: Please make sure to read this page on the WeBWorK policies for this class.

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Don’t forget to include the problem itself in your question, as that will make it easier for you to get a quick response!