Copyright © Cengage Learning. All rights reserved. Real Numbers and Their Basic Properties 1

Copyright © Cengage Learning. All rights reserved. Section 1.3 Exponents and Order of Operations

3 1. Identify the base and the exponent to simplify an exponential expression. 2. Evaluate a numeric expression following the order of operations. Use the correct geometric formula for an application. Objectives

4 Identify the base and the exponent to simplify an exponential expression 1.

5 Base and Exponent To show how many times a number is to be used as a factor in a product, we use exponents. 2 3 = 2  2  2 = 8 In the expression 2 3, 2 is called the base and 3 is called the exponent. The exponent of 3 indicates that the base of 2 is to be used as a factor three times: 2 3 = 2  2  2 = 8 3 factors of 2

6 Comment Note that 2 3 = 8. This is not the same as 2  3 = 6. In the expression x 5 (called an exponential expression or a power of x), x is the base and 5 is the exponent. The exponent of 5 indicates that a base of x is to be used as a factor five times. x 5 = x  x  x  x  x 5 factors of x Base and Exponent

7 In expressions such as 7, x, or y, the exponent is understood to be 1: 7 = 7 1 x = x 1 y = y 1 In general, we have the following definition. Natural-Number Exponents If n is a natural number, then x n = x  x  x ...  x n factors of x Exponent and Base

8 Example 1 Write each expression without exponents. a. 4 2 = 4  4 b. 5 3 = 5  5  5 c. 6 4 = 6  6  6  6 d. Read 4 2 as “4 squared” or as “4 to the second power.” Read 5 3 as “5 cubed” or as “5 to the third power.” Read 6 4 as “6 to the fourth power.” Read as “ to the fifth power.” = 16 = 125 = 1,296

9 We can find powers using a calculator. For example, to find , we enter these numbers and press these keys: Either way, the display will read. Some scientific calculators have an key rather than a key. Using a scientific calculator Using a graphing calculator Base and Exponent

10 Evaluate a numeric expression following the order of operations 2.

11 Order of Operations We consider the expression  4, which contains the operations of addition and multiplication. We can calculate this expression in two different ways. We can perform the multiplication first and then perform the addition. Or we can perform the addition first and then perform the multiplication. However, we will get different results. Multiply first Add first  4 =  4 = 5  4 = 14 = 20 Multiply 3 and 4. Add 2 and 12. Add 2 and 3. Multiply 5 and 4. Different results

12 Order of Operations To eliminate the possibility of getting different answers, we will agree to perform multiplications before additions. The correct calculation of  4 is  4 = = 14 To indicate that additions are to be done before multiplications, we use grouping symbols such as parentheses ( ), brackets [ ], or braces { }. Do the multiplication first.

13 The operational symbols and fraction bars are also grouping symbols. In the expression (2 + 3)4, the parentheses indicate that the addition is to be done first: (2 + 3)4 = 5  4 = 20 To guarantee that calculations will have one correct result, we will always perform calculations in the following order. Do the addition within the parentheses first. Order of Operations

14 Rules for the Order of Operations Use the following steps to perform all calculations within each pair of grouping symbols, working from the innermost pair to the outermost pair. 1. exponential expressions. 2. multiplications and divisions, working from left to right. 3. additions and subtractions, working from left to right. Order of Operations

15 4. Because a fraction bar is a grouping symbol, simplify the numerator and the denominator in a fraction separately. Then simplify the fraction, whenever possible. Comment Note that 4(2) 3  (4  2) 3 : 4(2) 3 = 4  2  2  2 = 4(8) = 32 and (4  2) 3 = 8 3 = 8  8  8 = 512 Likewise, 4x 3  (4x) 3 because 4x 3 = 4xxx and (4x) 3 = (4x)(4x)(4x) = 64xxx Order of Operations

16 Example 3 Evaluate: (8 – 3  2) Solution: We perform the work within the parentheses first and then simplify (8 – 3  2) = (8 – 6) = (2) = (2) Do the multiplication within the parentheses. Find the value of the exponential expression. Do the subtraction within the parentheses.

17 Example 3 – Solution = = 129 Do the multiplication. Do the addition. cont’d

18 Use the correct geometric formula for an application 3.

19 Example 6 – Circles Use the information in Figure 1-17 to find: a. the circumference b. the area of the circle. Figure 1-17

20 Example 6(a) – Solution The formula for the circumference of a circle is C =  D where C is the circumference,  can be approximated by, and D is the diameter—a line segment that passes through the center of the circle and joins two points on the circle. We can approximate the circumference by substituting for  and 14 for D in the formula and simplifying. C =  D

21 Example 6(a) – Solution C   14 C  C  44 The circumference is approximately 44 centimeters. Read  as “is approximately equal to.” Multiply the fractions and simplify. cont’d

22 To use a calculator, we enter these numbers and press these keys: Either way, the display will read The result is not 44, because a calculator uses a better approximation for  than. Using a scientific calculator Using a graphing calculator cont’d Example 6(a) – Solution

23 Comment A segment drawn from the center of a circle to a point on the circle is called a radius. Since the diameter D of a circle is twice as long as its radius r, we have D = 2r. If we substitute 2r for D in the formula C =  D, we obtain an alternate formula for the circumference of a circle: C = 2  r. cont’d Example 6(a) – Solution

24 The formula for the area of a circle is A =  r 2 where A is the area,  , and r is the radius of the circle. We can approximate the area by substituting for  and 7 for r in the formula and simplifying. A =  r 2 Evaluate the exponential expression. cont’d Example 6(a) – Solution

25 The area is approximately 154 square centimeters. To use a calculator, we enter these numbers and press these keys: The display will read Multiply the fractions and simplify. Using a scientific calculator Using a graphing calculator cont’d Example 6(a) – Solution

26 Table 1-1 shows the formulas for the perimeter and area of several geometric figures. Table 1-1 Use the correct geometric formula for an application

27 The volume of a three-dimensional geometric solid is the amount of space it encloses. Table 1-2 shows the formulas for the volume of several solids. Table 1-2 Use the correct geometric formula for an application

28 When working with geometric figures, measurements are often given in linear units such as feet (ft), centimeters (cm), or meters (m). If the dimensions of a two-dimensional geometric figure are given in feet, we can calculate its perimeter by finding the sum of the lengths of its sides. This sum will be in feet. If we calculate the area of a two-dimensional figure, the result will be in square units. Use the correct geometric formula for an application

29 For example, if we calculate the area of the figure whose sides are measured in centimeters, the result will be in square centimeters (cm 2 ). If we calculate the volume of a three-dimensional figure, the result will be in cubic units. For example, the volume of a three-dimensional geometric figure whose sides are measured in meters will be in cubic meters (m 3 ). Use the correct geometric formula for an application