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# DrDelMath

## Chapter 5: Exponents and Scientific Notation Chapter 7: Rational Expressions

#### Section 5.5a: Exponents --- Section 5.1 in the text and the beginning of Section 5.5

Page 315: Vocabulary: 1 - 6.
Page 316:Exercises: 3, 5, 19, 41,119, 121.
Page 350: Vocabulary: 3, 4.
Page 350: Exercises: 7, 15, 17.
Page 557: Exercises 5, 9, 10, 41.
You should be able to write: 1) Product rule for exponents. 2) Power rule for exponents. 3) Power of a product rule. 4) Power of a quotient rule. 5) Quotient rule for exponents. 6) Zero Exponent Rule.
Page 345: You must be able to write the definition of negative exponent as stated at the top of the page and its main consequence at the bottom of the page.
Suplemental Exercises:A3, B4, E2. Solutions HERE
Suplemental Exercises:1b, 3c, 5f, 6c, 7d, 7j, 8. Solutions HERE
Suplemental Exercises:32, 36, 37. Solutions HERE
Concept 1: Is $\sqrt{83}$ a:

1. Mathematics object? If yes what kind of object is it? Be specific.
2. Binary Relation? If yes what relation is it? Be specific.
3. Unary Relation? If yes what relation is it? Be specific.
4. Binary Operation? If yes what operation is it? Be specific.

#### Section 5.5b: Scientific Notation

Here are some common uses of scientific notation.
1 light year is 9.4605284 X 1015 meters.
Avogadro's number is the number of molecules in a mole. It is 6.0221415 × 1023
Boltzmann constant =1.3806488 × 10-23
The Bohr radius 0.529×10-10 m is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state.

### Addition

Numbers written in scientific notation are similar to terms and in computations they behave like terms. Coefficient and exponent have the same meaning. The base 10 plays the same role in scientific notations as the variable base does in a term.

Compare the definition for product of numbers written in scientific notation with the definition of product of terms. Then compare a few computational examples of multiplying terms with computational examples of multiplying numbers written in scientific notation.
Do similar comparisions of the process of division.

The concept of like terms is introduced to facilitate addition (commonly called combining) of like terms.
In a similar manner we can define like scientific numbers to be two numbers written as coefficients times the same powers of 10.
NOTE:This is a stipulative definition. I am stipulating that is what "like scientific numbers" will mean in this discussion.

The above definition of addition (combining) like terms is really not necessary because the process is an immediate consequence of the Distributive Property.
For example the Distributive Property directs us to factor out x5 to perform the following addition: 3x5 + 5x5 = (3 + 5)x5 = 8x5.

In a similar fashion we can use the Distributive Property to add like scientific numbers. Consider the following examples.
Observe that the sum is not always a number written in scientific notation. Of course the sum can be converted to scientific notation or standard notation.

After a few examples it becomes clear that this manner of performing addition really does not depend on the numbers being written in scientific nootation. It depends only on the Distributive Property.
As a result we are able to extend the technique to all sorts of additions.

Page 350: Exercises: 69, 71, 112, 113, 119.
Page 316:Exercises:11, 29, 33, 37, 119.
Page 557: Exercises 2, 6, 12, 42.
Page 347: You must be able to write the definition of scientific notation as stated at the bottom of the page.
Page 854: Exercises 1, 7, 32, 45.
Suplemental Exercises:3, 8. Solutions HERE
Write The Law of Trichotomy.
Write The Zero Factor Property.
Write the definition of opposite of a real number.

#### Section 7.0: Essential Prerequisites

Everything in this chapter can be summarized in the following three sentences.
Rational numbers (common fractions) are ratios of integers.
Rational expressions are ratios of polynomials.
The rules for operating (adding, subtracting, multiplying, and dividing) with rational expressions are the same as operating with rational numbers (common fractions).

Division of fractions is exactly like all divisions in mathematics

Because the multiplicative inverse of a fraction is its reciprocal, the conversion to multiplication looks like:

After converting to multiplication and performing the multiplication according to the definition of multiplication we get the following rule.

All of this boils down to a fairly simple rule for working with division of fractions as set forth in the following definition.

Examples of Division of FractionsHERE

Subtraction of fractions is exactly like all subtractions in mathematics

Using one of the four forms for the opposite of a fraction makes the conversion to addition look like:

From which we get the following definition of subtraction for fractions.

Page 22: Vocabulary: 1 - 6.
Page 23: Exercises:15, 28, 30, 47, 51, 53.
Page 316:Exercises:55, 65, 75, 79.
Page 350: Exercises: 75, 77, 93, 97, 111.
Give an example of a whole number which is not a natural number.
How many real numbers are in the interval [5, 9)?
The base of a triangle is 24 inches long. The area of the triangle is 60 square inches. What is the height of the triangle?
The length of one of the bases of a trapezoid is one more than twice that of the other base. The altitude is 2 in. If the area of the trapezoid is 19, what are the lengths of its bases?
See Examples 5 and 7 in Geometry Examples

#### Section 7.1: Simplifying Rational Expressions

Examples:
The following are rational expressions because each is the quotient of two polynomials.

$\begin{array}{l}\frac{x-2}{3x+1},\frac{5{x}^{3}+2x-3}{4x+1},\frac{3}{x+8}\\ \\ \frac{{x}^{5}+3{x}^{2}+1}{7},\genfrac{}{}{0.2ex}{}{\frac{3x-5}{x+6}}{\frac{8-9x}{14}}\end{array}$

Examples:
The following are not rational expressions because none is the quotient of two polynomials.

$\frac{x-2}{\sqrt{3x+1}}$ , $\frac{5{x}^{2}-3{x}^{-4}+1}{4x+11}$ , $\frac{{3}^{x}}{2x+9}$

Examples:
The domain of $\frac{x+2}{x-3}$ is $\left(-\infty ,3\right)\cup \left(3,\infty \right)$

The domain of $\frac{2}{4x-20}$ is $\left(-\infty ,5\right)\cup \left(5,\infty \right)$

The domain of $\frac{{x}^{3}}{2x-9}$ is $\left(-\infty ,\frac{9}{2}\right)\cup \left(\frac{9}{2},\infty \right)$

The domain of $\frac{3x+1}{\left(x-3\right)\left(x+2\right)}$ is $\left(-\infty ,-2\right)\cup \left(-2,3\right)\cup \left(3,\infty \right)$

The Fundamental Property of Fractions is the primary tool for simplifing fractions and rational expressions.

Study Directive:
Study the following in your textbook. Beginning and Intermediate Algebra Textbook by Elayn Martin-Gay. Chapter 7: Section 7.1: Objective 2: Simplfying Rational Expressions Pages 441 - 445.

Pay close attention to the reasons she presents (in blue type) for almost every step in the examples. As you study this material relate each process and example to what you already know about fractions.

At the top of Page 442 there are two excellent visualizations of the "common factor of 1". You should strive to mentally "see" this visual each time you use the process commonly called cancelling. Unless you rely on that visualization when cancelling, you will make errors.

Cancelling is so frequently abused that your author very wisely does not use that terminology in this discussion. Instead she gets right at the mathematical heart of the process -- removing factors of 1. After you think you have mastered everything for this objective, go back through all of it. This time you should observe that removing a factor of 1 from a product is the only thing discussed. Since you have long been familiar with the idea of not writing the 1 when it is a factor, there really is nothing new in this objective. That should make it easier.

Examples of Simplifying Rational Expressions HERE

Page 447: Vocabulary: 1 - 8.
Page 447: Exercises:1, 3, 5, 15, 29, 61.
Page 316:Exercises: 41, 59, 63, 71, 77.
Page 350: Exercises: 83, 85, 87, 103.
Page 23: Exercises: 21, 29, 31, 49, 53, 61, 62.
Page 557: Exercises 1, 11, 18, 45.
Supplemental Exercises:4, 6. Solutions HERE

#### Section 7.2: Multiplying and Dividing Rational Expressions

Fundamental Property of Fractions: If a represents a real number and b and c represent non-zero real numbers then $\frac{{ac}}{{bc}} = \frac{a}{b}$

Examples of Multiplying Rational Expressions HERE
Examples of Division of Rational Equations HERE

Page 458: Exercises: 1, 5, 7, 11, 13, 17, 19, 21, 23, 27, 29.
Page 316:Exercises: 55, 65, 75, 79, 121.
Page 350: Exercises:87, 97, 99,101, 103, 119
Page 447: Exercises:1, 5, 9, 11, 17, 27, 29, 41.
Page 557: Exercises 4, 32, 43, 67.
Suplemental Exercises:B3, C2, E4, F4. Solutions HERE

#### Section 7.3: Adding and Subtracting Rational Expressions with the Same Denominators

Fundamental Property of Fractions: If a represents a real number and b and c represent non-zero real numbers then $\frac{{ac}}{{bc}} = \frac{a}{b}$

Examples of Adding Rational Expressions with the Same Denominators HERE
Examples of Subtraction of Rational Expressions with the Same Denominators HERE

Page 466: Exercises: 1, 5, 7, 29, 45, 82.
Page 316:Exercises: 3, 5, 19, 121.
Page 350: Exercises: 69, 71, 111, 113.
Page 447: Exercises: 3, 11, 13, 21, 31, 64.
Page 458: Exercises: 1, 5, 7, 13, 19, 21, 23.
Page 557: Exercises 56, 65, 70.
Page 23: Exercises: 29, 47, 55, 57, 63.

#### Section 7.4: Adding and Subtracting Rational Expressions with Different Denominators

Fundamental Property of Fractions: If a represents a real number and b and c represent non-zero real numbers then $\frac{{ac}}{{bc}} = \frac{a}{b}$

Examples of Expanding Rational Expressions HERE

Examples of Reducing Rational Expressions HERE

Examples of Adding Rational Expressions with Different Denominators HERE
Examples of Subtraction of Rational Expressions with Different Denominators HERE

Page 472: Exercises: 1, 5, 15, 17, 19, 41, 43, 59, 81.
Page 350: Exercises: 99,101, 112, 119.
Page 23: Exercises: 29, 30, 31, 49, 63.
Page 447: Exercises: 11, 15, 61.
Page 458: Exercises: 1, 5, 7, 11, 13, 17, 19, 21, 23, 27, 29.
Page 466: Exercises: 17, 21, 25, 33, 41, 50, 77, 79.
Suplemental Exercises:34, 35, 37. Solutions HERE

#### Section 7.5: Solving Equations containing Rational Expressions

Recall that real numbers are polynomials, they are called constant polynomials.

In this section when we speak of solving rational equations, the reference is to rational equations which are not polynomial equations. We are concerned here with equations that have variables in the denominators. To solve that kind of equation we must be aware of the following VERY important facts.

Examples of working with rational expressions and rational equationsHERE
Examples of Solving Linear Equations and Solving Rational EquationsHERE

Page 479: Exercises: 1, 7, 15, 37, 41, 51.
Page 316:Exercises: 11, 29, 33, 37, 55.
Page 350: Exercises: 97, 103, 111, 119
Page 447: Exercises: 17, 27, 29, 41.
Page 458: Exercises: 7, 17, 23, 29.
Page 466: Exercises: >11, 13, 23, 35, 58, 61, 75.
Page 472: Exercises: 9, 11, 47, 82.
Suplemental Exercises:1d, 1e, 3e, 3j, 4c, 4d, 5h, 5k, 6e, 6h, 7i, 7d, 10, 12. Solutions HERE

#### Section 7.7: Simplifying Complex Fractions

IGNORE METHOD 2 ON PAGE 497. DO NOT USE IT.

ReminderYour primary goal is not to "get the answer" but rather to learn how various fundamental mathematics properties are used to construct stategies for analyzing/solving problems.
You should always avoid gimmics devised to "get the answer" for inane contrived textbook exercises.
Enhance your critical thinking skills by learning to work with concepts such as deductive reasoning, abstraction, generalization, methods of analyzing new problems, as well as some mathematical facts.

Examples of Simplifying Complex Rational Expressions HERE

Page 472: Exercises: 13, 17, 35, 49, 67.
Page 316:Exercises: 3, 33, 37, 41, 65.
Page 350: Exercises: 69.
Page 458: Exercises: 11, 19.
Page 472: Exercises: 20, 23, 55, 87.
Page 479: Exercises: 17, 18, 43, 49, 53.
Write all the subsets of {a, k, 5, 9}. There are 16. Get organized:Write the subsets with: no elelments, 1 element, 2 elements, 3 elements, and 4 elements