CHAPTER R Basic Concepts of Algebra R 1

CHAPTER R: Basic Concepts of Algebra R. 1 The Real-Number System R. 2 Integer Exponents, Scientific Notation, and Order of Operations R. 3 Addition, Subtraction, and Multiplication of Polynomials R. 4 Factoring R. 5 The Basics of Equation Solving R. 6 Rational Expressions R. 7 Radical Notation and Rational Exponents Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

R. 2 Integer Exponents, Scientific Notation, and Order of Operations · · · Simplify expressions with integer exponents. Solve problems using scientific notation. Use the rules for order of operations. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Integers as Exponents When a positive integer is used as an exponent, it indicates the number of times a factor appears in a product. For any positive integer n, where a is the base and n is the exponent. Example: 84 = 8 • 8 • 8 For any nonzero real number a and any integer m, a 0 = 1 and. Example: a) 80 = 1 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley b) Slide R. 2 - 4

Properties of Exponents Product rule Raising a product to a power (ab)m = ambm Quotient rule Raising a quotient to a power Power rule (am)n = amn Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 2 - 5

Examples – Simplify. a) r 2 • r 5 = r ( 2 + 5) = r 3 b) d) (3 a 3)4 = 34(a 3)4 = 81 a 12 or e) c) (p 6)4 = p ‒ 24 or Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 2 - 6

Scientific Notation Use scientific notation to name very large and very small positive numbers and to perform computations. Scientific notation for a number is an expression of the type N 10 m, where 1 N < 10, N is in decimal notation, and m is an integer. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 2 - 7

Examples Convert to scientific notation. a) 17, 432, 000 = 1. 7432 107 b) 0. 0000024 = 2. 4 10 10 Convert to decimal notation. a) 3. 481 106 = 3, 481, 000 b) 5. 874 10 5 = 0. 00005874 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 2 - 8

Another Example Chesapeake Bay Bridge-Tunnel. The 17. 6 -mile-long tunnel was completed in 1964. Construction costs were $210 million. Find the average cost per mile. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 2 - 9

Rules for Order of Operations 1. 2. 3. 4. Do all calculations within grouping symbols before operations outside. When nested grouping symbols are present, work from the inside out. Evaluate all exponential expressions. Do all multiplications and divisions in order from left to right. Do all additions and subtractions in order from left to right. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 2 - 10