1. Key Stone Problem… Key Stone Problem… next Set 8 © 2007 Herbert I. Gross

2. You will soon be assigned problems to test whether you have internalized the material in Lesson 8 of our algebra course. The Keystone Illustration below is a prototype of the problems you'll be doing. Work out the problem on your own. Afterwards, study the detailed solution we've provided. Instructions for the Keystone Problems next © 2007 Herbert I. Gross

3. As a teacher/trainer, it is important for you to understand and be able to respond in different ways to the different ways individual students learn. The more ways you are ready to explain a problem, the better the chances are that the students will come to understand. next © 2007 Herbert I. Gross

4. The problem we have chosen not only reinforces the notion of scientific notation, but it also provides a review of several of the properties of the arithmetic of exponents. next © 2007 Herbert I. Gross Note

5. next Problem Write the following number in scientific notation: 2 × 3 × 7 × (2 × 10 -6 ) 2 × (5 × 10 4 ) 3 10 -3 Keystone Illustrations for Lesson 8 next Answer: 2.1 × 10 7 © 2007 Herbert I. Gross

6. Solution for the Problem We may rewrite the problem in the form… next © 2007 Herbert I. Gross 2 × 3 × 7 × (2 × 10 -6 ) 2 × (5 × 10 4 ) 3 ÷ 10 -3 next 2 × 3 × 7 × (2 × 10 -6 ) 2 × (5 × 10 4 ) 3 10 -3

7. Solution for the Problem To eliminate the parentheses we use the property that… (b × c) n = b n × c n next © 2007 Herbert I. Gross (2 1 × 10 -6 ) 2 = 2 1(2) × 10 -6(2) = 2 2 × 10 -12 next (5 × 10 4 ) 3 = 5 1(3) × 10 4(3) = 5 3 × 10 12 Thus… and… next

8. Solution for the Problem If we now replace the parenthetical expressions by their values, we see that the desired number is… next © 2007 Herbert I. Gross next 2 × 3 × 7 × ( 2 2 × 10 -12 ) × ( 5 3 × 10 12 ) ÷ 10 -3 2 × 3 × 7 × (2 × 10 -6 ) 2 × (5 × 10 4 ) 3 ÷ 10 -3

9. Solution for the Problem Dividing by 10 -3 is the same as multiplying by the reciprocal of 10 3, and since the reciprocal of 10 -3 is 10 3, next © 2007 Herbert I. Gross next 2 × 3 × 7 × 2 2 × 10 12 × 5 3 × 10 12 × 10 3 2 × 3 × 7 × 2 2 × 10 12 × 5 3 × 10 12 ÷ 10 -3 we may rewrite the expression as…

10. Solution for the Problem Since the computation involves only multiplication we can regroup the factors in any way that we wish. next © 2007 Herbert I. Gross next 3 × 7 × 2 × 2 2 × 5 3 × 10 -12 × 10 12 × 10 3 2 × 3 × 7 × 2 2 × 10 12 × 5 3 × 10 12 × 10 3 In particular, we may rewrite the expression as…

11. Solution for the Problem And since 2 × 2 2 = 2 3 we may rewrite the expression as… next © 2007 Herbert I. Gross next If we next observe that 2 3 × 5 3 = (2 × 5) 3 = 10 3, we may rewrite the expression as… 3 × 7 × (10 3 × 10 -12 × 10 12 × 10 3 ) 3 × 7 × (2 3 × 5 3 ) × 10 -12 × 10 12 × 10 3 3 × 7 × 2 × 2 2 × 5 3 × 10 -12 × 10 12 × 10 3 next

12. Solution for the Problem By our rule for multiplying like bases, we know that… 10 3 × 10 -12 × 10 12 × 10 3 = 10 3+ -12 + 12 + 3 = 10 6. next © 2007 Herbert I. Gross next Therefore, we may rewrite the expression in the equivalent form… 21 × 10 6 3 × 7 next × 10 6 3 × 7 × (10 3 × 10 -12 × 10 12 × 10 3 )

13. Solution for the Problem While 21 × 10 6 is numerically correct, it is not in scientific notation because 21 is not between 1 and 10. However, we may rewrite 21 in the form 21 = 2.1 × 10 ( = 2.1 × 10 1 ), in which case the expression becomes… next © 2007 Herbert I. Gross next 2.1 × 10 7 2.1 × (10 1 × 10 6 ) (2.1 × 10 1 ) × 10 6 21 × 10 6

14. Remember that from a mathematical point of view, 2.1 × 10 7, 21 × 10 6, and 210 × 10 5, etc. all name the same number, but only 2.1 × 10 7 is in scientific notation. next © 2007 Herbert I. Gross Note In terms of significant figures that we discussed in the lesson, 2.1 × 10 7 would mean that the measurement had 2 significant figures. That is, the exact value would be between 2.05 × 10 7 and 2.15 × 10 7. next

15. And if we wanted to emphasize that the measurement had 3 significant figures, we would have written 2.10 × 10 7, which would tell us that the exact value would be between 2.095 × 10 7 and 2.105 × 10 7. However when measurements are not involved, 2.1 × 10 7, 2.10 × 10 7, and 2.100 × 10 7, etc. all name the number 21,000,000. next © 2007 Herbert I. Gross Note

16. As we mentioned in our lesson, architects often used such measurements as 2.1± 0.005 inches rather 2.10 inches than to indicate that the exact measurement was between 2.095 and 2.105. This method is less subject to misinterpretation than the use of significant figures. With the prevalence of calculators, it is not difficult to compute using such measurements as 2.1 ± 0.003, etc. next © 2007 Herbert I. Gross Note