ELEMENTARY & INTERMEDIATE ALGEBRA
These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: This material is extensively elaborated upon in my optional e-book GOLDen Mathematics: Intermediate Algebra. This section of material only is downloadable for nominal fee at www.lulu.com/content/378810. See "Tell me more about Keely's GM book".
Roots and Radicals
GOLDen Mathematics - Intermediate Algebra: Section 8.1
Supplemental Sites: MathOL Links - Alg
8.1
Topics of Importance
Radical expressions: definition, terminology
Evaluate roots: perfect by hand, approx by calc
Radical functions: evaluate, graph, domain, range
Odd vs. even nth roots of xpower
Comments and Cautions
This chapter introduces roots like square roots, cube roots, etc., evaluating radicals on your calculator, evaluating radical functions,
finding domains of radical functions, and simplifying perfect radicals. After
9.2 we’ll start to work with radical expressions (like adding, multiplying, FOILing,
etc.), so build a strong foundation of the basics now! Here are
a couple of review rules to remember:
 | It is OK to have a negative number under an odd-indexed root. The answer to it will always be a negative number. For instance the cube root of -8 is -2. |
| It is not OK to have a negative number under an even-indexed root. For instance the fourth root of -16 is "not a Real number". |
Besides working arithmetic roots by hand and on a calculator (see my online Calculator Guide: Basic Arithmetic), roots of algebraic expressions get introduced too. One important rule:
| The answer from an even-indexed root has to be a positive number. For instance the square root of x2 can't be x since x might be negative. Instead we need absolute value bars to assure the answer is positive. So sqrt(x2)=|x|. |
Now try putting these rules together to determine when absolute values are needed on the answer and when they aren't. Only one of sqrt(x4), cubert(x6), sqrt(x6), cubert(x9) needs absolute values on the answer. Which one and why? (Let's discuss in class.)
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 10.1.)
| ch 10.1 pg 639 Note the definition of the "principle square root". Be aware that some text/websites will not only give the "principle" root but might discuss the "two square roots of a number". Personally I appreciate that the Blitzer text sticks to the "principle" root. But just in case you run into it in other sources, the "two square roots of a number" means, e.g., 9 has two square roots 3 and -3 since either squared would make 9. Whereas the "principle square root" occurs when the radical sign is already around the number and the answer is only positive, e.g., Ö9 = 3. |
| ch 10.1 At the bottom of page 639 and mid page 645 note that an an even root of a negative number is "not a Real number". Later in this chapter we will learn about "imaginary numbers" and then we will be able to get a result for the square root of a negative number such as Ö-9, but for now, you can just say that Ö-9 is a "non-Real number". |
| ch 10.1 pg 642-643 plus the box on page 646. Pay special attention to when you need the absolute value on the answer from a radical expression and when you don't. We will discuss this more in class.9.1 examples 14-15 You do not need to know how to produce on your grapher a curve from given data. The only thing you should know from these examples is if you are given the graphed data points, do the dots roughly make a shape that could be approximated by a radical function? I.e., do the dots take on a radical function shape? Actually "modeling" the data by finding a specific equation that would "fit" the data will be covered in a college level math course. |
Fractional Exponents
GOLDen Mathematics - Intermediate Algebra: Section 8.2
Supplemental Sites: MathOL Links - Alg
8.2
Topics of Importance
Convert fractional exponent expressions to radicals
Evaluate fractional exponent expressions: exact by hand, approx by calc
Rules of fractional exponents: product, power, quotient
Simplify fractional exponent expressions including FOIL
Simplify radicals via fractional exponents
Perform operations on fractional exponent expressions that do not contain
variables
Comments and Cautions
You know about positive whole number powers like x3
and negative integer powers like x-2 (recall x-2 means 1/x2).
Today we will learn about fractional powers like x½ and decimal
powers like x1.2. Variables to fractional powers can be rewritten as
radicals. Working with fractional powers is actually more efficient than
radicals. In fact some radical expressions cannot even be simplified without
using fractional exponents.
To convert from fractional exponents to radicals, the denominator of the fraction becomes the index of the root. The numerator of the fraction remains as the power on the variable. So x2/3=cubert(x2) and x½=sqrt(x). So as the chapter continues, sometimes the expressions will be written as radicals, sometimes as fractional exponents. All the rules of exponents that you learned with integer powers hold for fractional powers. E.g., (x2/3)1/2=x2/3·1/2=x1/3. This is a very important section that is common to stumble over, so practice lots of problems in this section, they are quickies and worth trying a vast variety.
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 10.2.)
| ch 10.2 The definitions on pg 651 and 653 describe how to convert between expressions with fractional exponents and radical form. It is important to be able to do this to and fro. |
| ch 10.2 pg 656 The "laws/properties of fractional/rational exponents" are important not just to memorize but to know how and when to use each! |
Radical Expression Operations
GOLDen Mathematics - Intermediate Algebra: Section 8.3
Supplemental Sites: MathOL Links - Alg
8.3
Topics of Importance
Radical Expression ops: simplify, multiply, divide,
add/subtract, distribute, FOIL
Rationalize the denominator
Factor by difference of squares using radicals
Comments and Cautions
In this section we learn to perform operations on and
manipulate radical expressions. These skills will primarily be useful when we
solve radical equations and application problems in future lessons. A
caution and some rules to note as you work through the material.
CAUTION: sqrt(4x2-9) is not equal to sqrt(4x2) - sqrt(9). You cannot break up square roots when the radicand is involved in addition or subtraction. But, if the 4x2 and 9 were multiplied together you could take the square root of each. I.e., sqrt(4x2·9) does equal sqrt(4x2)·sqrt(9) = 2x·3 = 6x. (Well, technically 2|x|·3 = 6|x| unless we were given that x≥0.)
RULE when multiplying radicals: When multiplying radicals that have the same index, you can rewrite them as one big radical with the radicands multiplied together underneath, and then simplify completely.
RULE when dividing radicals: When taking a root of a fraction, you can split it up into the root of the numerator over the root of the denominator. Its a good idea to reduce the fraction first though! On the other hand, when dividing roots that have the same index, you can rewrite them as one big root with the radicands divided as one fraction underneath, and then simplify completely. So sometimes you want to take the radical of a fraction and write it as a fraction of radicals and sometimes you want to take the fraction of radicals and write it as a radical of a fraction!
RULE when adding radicals: You can only add or subtract "like" radical expressions, meaning that they must have the same index and identical radicands.
RULE regarding rationalizing: Never leave a fraction under a radical nor a radical in the denominator of a fraction. It is illegal! Must "rationalize the denominator". This is an important process. Concentrate on rationalizing the denominator when the fraction has a square root in the bottom, but try some with higher roots too.
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 10.3-10.5.)
| ch 10.3-10.5 contain many important rules and processes. Spend ample time working through these to get them down well! Ch 10.5 even covers FOILing binomials that contain radical terms. Hmmm, could factoring be far behind? |
| ch 10.5 pg 680-684 covers "rationalizing the denominator" which is an important concept to learn. Particularly concentrate on those that have single terms in the denominator. Watch the index carefully as it somewhat complicates the process! |
| ch 10.5 pg 684-695 including example 7. SKIP all the "rationalizing the numerator" stuff as it is only used in calculus and even then only rarely. |
Complex Numbers
GOLDen Mathematics - Intermediate Algebra: Section 8.4
Supplemental Sites: MathOL Links - Alg
8.4
Topics of Importance
Terminology: complex number system, imaginary no., definition
of i, real and imaginary part, conjugate
Evaluate expressions involving sq roots of neg numbers
Equivalent complex numbers
Simplify i to a power
Graph on the complex plane
Operations with complex numbers (by hand and on calc)
Factor a sum of squares
Comments and Cautions
I am so excited to finally get to talk about imaginary
numbers! So far you have been constrained in the Real realm, but today your
world will open up to the Complex realm which includes imaginary numbers :)
There is plenty of new terminology to learn, algebraic processes, and calculator
processes. Be sure to read my online
Calculator Guide: Complex Numbers. The assignments newsgroup this week will
be focused on complex numbers, so plan to spend some time there.
Two cautions to take note of: Never leave i to a power. Never leave i in the denominator of a fraction. Be sure you know how to simplify each of these cases!
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 10.7.)
| ch 10.7 This entire section is important. Lots of new fun stuff to learn! See the boards for some mini-lectures, examples, and discussions :) |
Originally written: 2006-006-15
Last revision:
2008-02-02 06:35 PM