Exponents and the Order of Operations

0.3 Exponents and the Order of Operations 0.3 OBJECTIVES 1. Write a product of factors in exponential form 2. Evaluate an expression involving several operations Often in mathematics we define symbols that allow us to write a mathematical statement in a more compact or shorthand form. This is an idea that you have encountered before. For example, the repeated addition: 5 5 5 NOTE 5 5 5 15 and 3 5 15 can be rewritten as 3 5 Thus multiplication is shorthand for repeated addition. In algebra, we frequently have a number or variable that is repeated as a factor in an expression several times. For instance, we might have 5 5 5 NOTE A factor is a number or a variable that is being multiplied by another number or variable. To abbreviate this product, we write 5 5 5 5 3 This is called exponential notation or exponential form. The exponent or power, here 3, indicates the number of times that the factor or base, here 5, appears in a product. Exponent or power CAUTION Be careful: 5 3 is not the same as 5 3. Notice that 5 3 5 5 5 125 and 5 3 15. 5 5 5 5 3 Factor or base This is read 5 to the third power or 5 cubed. Example 1 Writing in Exponential Form Write 3 3 3 3, using exponential form. The number 3 appears 4 times in the product, so Four factors of 3 3 3 3 3 3 4 This is read 3 to the fourth power. CHECK YOURSELF 1 Rewrite each using exponential form. (a) 4 4 4 4 4 4 (b) 7 7 7 7 29

30 CHAPTER 0 AN ARITHMETIC REVIEW CAUTION Only one of these results can be correct. To evaluate an arithmetic expression, you need to know the order in which the operations are done. To see why, simplify the expression 5 2 3. Method 1 or Method 2 5 2 3 5 2 3 Add first. Multiply first. 7 3 5 6 21 11 Because we get different answers depending on how we do the problem, the language of mathematics would not be clear if there were no agreement on which method is correct. The following rules tell us the order in which operations should be done. Step by Step: The Order of Operations NOTE Parentheses and brackets are both grouping symbols. Later we will see that fraction bars and radicals are also grouping symbols. Step 1 Step 2 Step 3 Step 4 Evaluate all expressions inside grouping symbols first. Evaluate all expressions involving exponents. Do any multiplication or division in order, working from left to right. Do any addition or subtraction in order, working from left to right. Example 2 Evaluating Expressions NOTE Method 2 shown above is the correct one. Evaluate 5 2 3. There are no parentheses or exponents, so start with step 3: First multiply and then add. 5 2 3 5 6 11 Multiply first. Then add. CHECK YOURSELF 2 Evaluate the following expressions. (a) 20 3 4 (b) 9 6 3 When there are no parentheses, evaluate the exponents first. Example 3 Evaluating Expressions Evaluate 5 3 2. 5 3 2 5 9 Evaluate the power first. 45 CHECK YOURSELF 3 Evaluate 4 2 4.

EXPONENTS AND THE ORDER OF OPERATIONS SECTION 0.3 31 Both scientific and graphing calculators correctly interpret the order of operations. This is demonstrated in Example 4. Example 4 Using a Calculator to Evaluate Expressions Use your scientific or graphing calculator to evaluate each expression. Round the answer to the nearest tenth. (a) 24.3 6.2 3.53 When evaluating expressions by hand, you must consider the order of operations. In this case, the multiplication must be done first, then the addition. With a calculator, you need only enter the expression correctly. The calculator is programmed to follow the order of operations. Entering 24.3 6.2 3.53 ENTER NOTE With most graphing calculators, the final command is ENTER. With most other scientific calculators; the key is marked. yields the evaluation 46.186. Rounding to the nearest tenth, we have 46.2. (b) 2.45 3 49 8000 12.2 1.3 Some calculators use the carat (^) to designate powers. Others use the symbol x y (or y x ). Entering 2.45 ^ 3 49 8000 12.2 1.3 or 2.45 y x 3 49 8000 12.2 1.3 ENTER yields the evaluation 30.56. Rounding to the nearest tenth, we have 30.6. CHECK YOURSELF 4 Use your scientific or graphing calculator to evaluate each expression. (a) 67.89 4.7 12.7 (b) 4.3 55.5 3.75 3 8007 1600 Operations inside grouping symbols are done first. Example 5 Evaluating Expressions Evaluate (5 2) 3. Do the operation inside the parentheses as the first step. (5 2) 3 7 3 21 Add. CHECK YOURSELF 5 Evaluate 4 (9 3). The principle is the same when more than two levels of operations are involved.

32 CHAPTER 0 AN ARITHMETIC REVIEW Example 6 (a) Evaluate 4 (2 3) 3. Add inside the parentheses first. 4 (2 3) 3 4 (5) 3 Evaluate the power. 4 125 500 (b) Evaluate 5 (7 3) 2 10. Multiply. 5 (7 3) 2 10 5(4) 2 10 Evaluate the expression inside the parentheses. 5 16 10 Multiply. 80 10 70 Subtract. Evaluate the power. CHECK YOURSELF 6 Evaluate. (a) 4 3 3 8 11 (b) 12 4 (2 3) 2 CHECK YOURSELF ANSWERS 1. (a) 4 6 ; (b) 7 4 2. (a) 8; (b) 11 3. 64 4. (a) 8.2; (b) 190.92 5. 24 6. (a) 20; (b) 112

Name 0.3 Exercises Section Date Write each expression in exponential form. 1. 7 7 7 7 2. 2 2 2 2 2 2 3. 6 6 6 6 6 4. 4 4 4 4 4 4 4 5. 8 8 8 8 8 8 8 8 8 8 6. 10 10 10 7. 15 15 15 15 15 15 8. 31 31 31 31 31 31 31 31 31 31 Evaluate each of the following expressions. 9. 7 2 6 10. 10 4 2 11. (7 2) 6 12. (10 4) 2 13. 12 8 4 14. 10 20 5 15. (12 8) 4 16. (10 20) 5 17. 8 7 2 2 18. 48 8 4 2 ANSWERS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 19. 8 (7 2) 2 20. 48 (8 4) 2 21. 3 5 2 22. 5 2 3 23. (3 5) 2 24. (5 2) 3 25. 4 3 2 2 26. 3 2 4 8 21. 22. 23. 24. 25. 26. 33

ANSWERS 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 27. 7 (2 3 5) 28. 4 (3 2 7) 29. 3 2 4 6 2 30. 4 2 3 5 6 31. (2 4) 2 8 3 32. (3 2) 3 7 3 33. 4 (2 6) 2 34. 3 (8 4) 2 35. (4 2 6) 2 36. (3 8 4) 2 37. 3 (4 3) 2 38. 5 (4 2) 3 39. 3 4 3 2 40. 5 4 2 3 41. 4 (2 3) 2 25 42. 8 + 2 (3 + 3) 2 43. (4 2 3) 2 25 44. 8 (2 3 3) 2 Evaluate using your calculator. Round your answer to the nearest tenth. 45. (1.2) 3 2.0736 2.4 1.6935 2.4896 46. (5.21 3.14 6.2154) 5.12.45625 47. 1.23 3.169 2.05194 (5.128 3.15 10.1742) 48. 4.56 (2.34) 4 4.7896 6.93 27.5625 3.1269 (1.56) 2 49. Population doubling. Over the last 2000 years, the Earth s population has doubled approximately 5 times. Write this last factor in exponential form. 50. Volume of a cube. The volume of a cube with each edge of length 9 in. is given by 9 9 9. Write the volume using exponential notation. Insert grouping symbols in the proper place so that the given value of the expression is obtained. 51. 36 4 2 4; 2 52. 48 3 2 2 3; 2 53. 6 9 3 16 4 2; 29 54. 5 3 2 8 5 2; 28 50. 51. 52. 53. 54. Answers 1. 7 4 3. 6 5 5. 8 10 7. 15 6 9. 19 11. 54 13. 10 15. 1 17. 60 19. 144 21. 75 23. 225 25. 34 27. 21 29. 36 31. 40 33. 256 35. 196 37. 147 39. 21 41. 75 43. 96 45. 1.2 47. 7.8 49. 2 5 51. 36 (4 2) 4 53. (6 9) 3 (16 4) 2 34