P.1 Algebraic Expressions, Mathematical Models, and Real Numbers P.2 Exponents and Scientific Notation Objectives: Evaluate algebraic expressions, find intersection and unions of sets, simplify algebraic expressions, use properties of exponents and scientific notation. Algebraic Expressions Variable: A letter is used to represent various numbers. Algebraic Expression: A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots. Evaluating an algebraic expression means to find the value of the expression for a given value of the variable.. Copyright 2014, 2010, 2007 Pearson Education, Inc. 2 The Order of Operations Agreement 1. Perform operations within the innnermost parentheses and work outward. If the algebraic expression involves and fraction, treat the numerator and the denominator as if they were each enclosed in parentheses. 2. Evaluate all exponential expressions 3. Perform multiplication and divisions as they occur, working from left to right. 4. Perform additions and subtractions as they occur, working from left to right. Copyright 2014, 2010, 2007 Pearson Education, Inc. 3 1
Example: Evaluating an Algebraic Expression Evaluate 8 + 6(x 3) 2 for x = 13. Copyright 2014, 2010, 2007 Pearson Education, Inc. 4 Formulas and Mathematical Models Equation: Formed when an equal sign is placed between two algebraic expressions. Formula: Is an equation that uses variables to express a relationship between two or more quantities. Mathematical Modeling: The process of finding formulas to describe real-world phenomena. Copyright 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Using a Mathematical Model 2 The formula T = 4x + 341x+ 3194 models the average cost of tuition and fees, T, for public U.S. colleges for the school year ending x years after 2000. Use the formula to project the average cost of tuition and fees at public U.S. colleges for the school year ending in 2015. Copyright 2014, 2010, 2007 Pearson Education, Inc. 6 2
Sets Set is a collection of objects whose contents can be clearly determined. Elements: The objects in a set. We use braces, { }, to indicate that we are representing a set. {1, 2, 3, 4, 5,...} is an example of the roster method of representing a set. The three dots after the 5, called an ellipsis, indicates that there is no final element and that the listing goes on forever. Empty Set: If a set has no elements, or the null set, and is represented by the Greek letter phi,. Copyright 2014, 2010, 2007 Pearson Education, Inc. 7 Set-Builder Notation In set-builder notation, the elements of the set are described but not listed. xxis a counting number less than 6 { } is read, The set of all x such that x is a counting number less than 6. The same set written using the roster method is {1, 2, 3, 4, 5}. Copyright 2014, 2010, 2007 Pearson Education, Inc. 8 Definition of the Intersection of Sets The intersection of sets A and B, written A B, is the set of elements common to both set A and set B. This definition can be expressed in set-builder notation as follows: A B = { x x is an element of A and x is an element of B}. Copyright 2014, 2010, 2007 Pearson Education, Inc. 9 3
Example: Finding the Intersection of Two Sets Find the intersection: { 3,4,5,6,7 } { 3,7,8,9}. Copyright 2014, 2010, 2007 Pearson Education, Inc. 10 Definition of the Union of Sets The union of sets A and B, written AU B, is the set of elements that are members of set A or of set B or of both sets. This definition can be expressed in set-builder notation as follows: AU B= x x is an element of AOR x is an element of B. { } Copyright 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Finding the Union of Two Sets Find the union: { 3,4,5,6,7} U{ 3,7,8,9 }. Copyright 2014, 2010, 2007 Pearson Education, Inc. 12 4
The Set of Real Numbers The sets that make up the Real Numbers,, are: the set of Natural Numbers, the set of Whole Numbers, the set of Integers, {1,2,3,4,5,...} {0,1,2,3,4,5,...} {..., 5, 4, 3, 2, 1,0,1,2,3,4,5,...} the set of Rational Numbers, = a a and b are integers and b 0 b and the set of Irrational Numbers. Irrational numbers cannot be expressed as a quotient of integers. Copyright 2014, 2010, 2007 Pearson Education, Inc. 13 The Set of Real Numbers (continued) The set of real numbers is the set of numbers that are either rational or irrational: xxis rational or xis irrational. { } Copyright 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Recognizing Subsets of the Real Numbers Consider the following set of numbers: π { 9, 1.3,0,0.3,, 9, 10 }. 2 List the numbers in the set that are natural numbers. Copyright 2014, 2010, 2007 Pearson Education, Inc. 15 5
Absolute Value The absolute value of a real number a, denoted x the distance from 0 to a on the number line. The distance is always taken to be nonnegative., is Definition of absolute value Copyright 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Evaluating Absolute Value Rewrite the expression without absolute value bars: 2 1 answer: Rewrite the expression without absolute value bars: π 3 answer: Copyright 2014, 2010, 2007 Pearson Education, Inc. 17 Properties of Absolute Value Copyright 2014, 2010, 2007 Pearson Education, Inc. 18 6
Distance between Points on a Real Number Line If a and b are any two points on a real number line, then the distance between a and b is given by a b or b a. Copyright 2014, 2010, 2007 Pearson Education, Inc. 19 Properties of the Real Numbers The Commutative Property of Addition The Commutative Property of Multiplication The Associative Property of Addition The Associative Property of Multiplication Copyright 2014, 2010, 2007 Pearson Education, Inc. 20 Properties of the Real Numbers (continued) The Distributive Property of Multiplication over Addition The Identity Property of Addition The Identity Property of Multiplication Copyright 2014, 2010, 2007 Pearson Education, Inc. 21 7
Properties of the Real Numbers (continued) The Inverse Property of Addition The Inverse Property of Multiplication Copyright 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Simplifying an Algebraic Expression Simplify: 2 2 7(4x + 3 x) + 2(5 x + x). Copyright 2014, 2010, 2007 Pearson Education, Inc. 23 Properties of Negatives Copyright 2014, 2010, 2007 Pearson Education, Inc. 24 8
Properties of Exponents Copyright 2014, 2010, 2007 Pearson Education, Inc. 25 Properties of Exponents (continued) Copyright 2014, 2010, 2007 Pearson Education, Inc. 26 Properties of Exponents (continued) Copyright 2014, 2010, 2007 Pearson Education, Inc. 27 9
Properties of Exponents (continued) Copyright 2014, 2010, 2007 Pearson Education, Inc. 28 Simplifying Exponential Expressions An exponential expression is simplified when: No parentheses appear. No powers are raised to powers. Each base occurs only once. No negative or zero exponents appear. Copyright 2014, 2010, 2007 Pearson Education, Inc. 29 Example: Simplifying Exponential Expressions Simplify: 3 6 4 (2 xy). Simplify: 2 5x. 4 y Copyright 2014, 2010, 2007 Pearson Education, Inc. 30 10
Scientific Notation A number is written in scientific notation when it is expressed in the form a 10 n where the absolute value of a is greater than or equal to 1 and less than 10, (1 a < 10), and n is an integer. Copyright 2014, 2010, 2007 Pearson Education, Inc. 31 Example: Converting from Decimal Notation to Scientific Notation Write in scientific notation: 5,210,000,000 Write in scientific notation: 0.00000006893 Copyright 2014, 2010, 2007 Pearson Education, Inc. 32 Computations with Scientific Notation Properties of exponents are used to perform computations with numbers that are expressed in scientific notation. Copyright 2014, 2010, 2007 Pearson Education, Inc. 33 11
Example: Computations with Scientific Notation Perform the indicated computation, writing the answer in scientific notation: ( 7.1 10 5 )( 5 10 7 ) Copyright 2014, 2010, 2007 Pearson Education, Inc. 34 Example: Computations with Scientific Notation Perform the indicated computation, writing the answer in scientific notation: 6 1.2 10 3 3 10 Copyright 2014, 2010, 2007 Pearson Education, Inc. 35 12