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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 4 1.3 Exponents, Roots, and Order of Operations Objectives 1.Use exponents. 2.Identify exponents and bases. 3.Find square roots. 4.Use the order of operations. 5.Evaluate algebraic expressions for given values of variables.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 5 Using Exponents Factors are two or more numbers whose product is a third number. Exponents are a way of writing products of repeated factors. 1.3 Exponents, Roots, and Order of Operations Base Exponent 3 4, read as “3 to the fourth power”, uses 3 as a factor 4 times and equals 81.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 6 Using Exponents Exponential Expression If a is a real number and n is a natural number, 1.3 Exponents, Roots, and Order of Operations where n is the exponent, a is the base, and is an exponential expression. Exponents are also called powers.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 7 1.3 Exponents, Roots, and Order of Operations Using Exponential Notation Write each expression Using exponents: Exponential notation: 6 · 6 · 6 · 6 · 66565 (0.7) (0.7) (0.7) 40 m · m · mm3m3 ( –y) ( –y) (–y)4(–y)4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 8 1.3 Exponents, Roots, and Order of Operations Evaluating Exponential Expressions Evaluate the expression: Exponential notation: 7272 7 · 7 = 49 (0.2) 3 (0.2) (0.2) (0.2)= 0.008 m4m4 m · m · m · m (–4) 4 (–4) (–4) (–4) (–4) = 256

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 9 1.3 Exponents, Roots, and Order of Operations Tips to Remember The product of an even number of negative factors is positive. The product of an odd number of negative factors is negative. To raise a number to a power on a calculator, enter the following: E.g., 2 3 2 y x 3 = or 2 x y 3 =

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 10 1.3 Exponents, Roots, and Order of Operations Identifying Exponents and Bases Identify the Exponent and Base ExponentBase 11 2 211 –4 3 34 (–4) 4 4 –4 –(0.8) 5 50.8

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 11 1.3 Exponents, Roots, and Order of Operations Be Sure to Identify the Base Correctly CAUTION

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 12 Square Roots 1.3 Exponents, Roots, and Order of Operations Squaring a number and taking its square root are opposites. 64 has two square roots: 8 and –8. Principle (positive) square root of 64 is denoted with Negative square root of 64 is denoted with

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 14 Finding Square Roots 1.3 Exponents, Roots, and Order of Operations Each square root is given.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 15 Order of Operations 1.3 Exponents, Roots, and Order of Operations When an expression involves more than one operation symbol, use the following: 1.Work separately above and below any fraction bar. 2.If grouping symbols such as parentheses ( ), square brackets [ ], or absolute value bars | | are present, start with the innermost set and work outward. 3.Evaluate all powers, roots, and absolute values. 4.Do any multiplications or divisions in order, working from left to right. 5.Do any additions or subtractions in order, working from left to right.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 16 = 6 + 18 ÷ (– 3) 2 –7 + 4 6= –7 + 4 6 Evaluating Expressions 1.3 Exponents, Roots, and Order of Operations Simplify: 6 + 18 ÷ (– 3) 2 = –7 + 24 =17 = 6 + (–6) 2 = 6 + (–12) = –6

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 17 5 4 2 + 10 ÷ ( 8 – 6) Evaluating Expressions 1.3 Exponents, Roots, and Order of Operations Simplify: = 5 4 2 + 10 ÷ ( 8 – 6) = 5 4 2 + 10 ÷ 2 = 5 16 + 10 ÷ 2 = 80 + 10 ÷ 2 = 80 + 5 = 85

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 18 Evaluating Expressions 1.3 Exponents, Roots, and Order of Operations Simplify: 1 3 12 + (– 18 + 15 ÷ 3) 1 3 = 1 3 12 + (– 18 + 5)= 1 3 12 + (– 13) = 4 + (– 13)= –9=

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 20 Algebraic Expressions 1.3 Exponents, Roots, and Order of Operations Any collection of numbers, variables, operation symbols, and grouping symbols, such as is called an algebraic expression. Algebraic expressions have different numerical values for different values of the variables. and

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 21 Evaluating Expressions for Given Values of Variables 1.3 Exponents, Roots, and Order of Operations The cost for a season pass to a state park is \$12 per person. The amount of dollars a family of x members would pay can be represented by \$12x. Cost per person = \$12 Number of persons = x Total cost = \$12x 3 member family 5 member family Total cost = \$12x 12 3 = \$36 12 5 = \$60

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 22 Evaluating Expressions 1.3 Exponents, Roots, and Order of Operations If c = 4 and b = –3, evaluate the expression: 3c – 7b = 3(4) – 7(–3) = 12 + 21 = 33

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 23 Evaluating Expressions 1.3 Exponents, Roots, and Order of Operations r = – 1 s = 64 t = –7 Use parentheses to avoid errors. Given Substitute and evaluate.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 24 Evaluating Expressions 1.3 Exponents, Roots, and Order of Operations The price per gallon of gasoline can be approximated for the years 2006 – 2008 by substituting a given year for x in the expression 0.17 x – 338.07 and then evaluating. Approximate the price of a gallon of gas in the year 2007.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 25 Evaluating Expressions 1.3 Exponents, Roots, and Order of Operations The approximate price of a gallon of gas in the year 2007, rounded to the nearest cent is 0.17x – 338.07 = 0.17(2007) – 338.07 = 341.19 – 338.07 = \$3.12

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 26 Evaluating Expressions 1.3 Exponents, Roots, and Order of Operations We can create a table to show how the price of gas changed during these years. YearPrice Per Gallon 2006\$2.95 2007\$3.12 2008\$3.29