4-2 Negative Exponents

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1 1. Write each expression using a positive exponent. 5. Write each fraction as an expression using a negative exponent other than esolutions Manual - Powered by Cognero Page 1

2 9. When a baseball is hit, it comes in contact with the bat for less than of a second. Write using a negative exponent other than Evaluate each expression if x 4 and y Write each expression using a positive exponent. esolutions Manual - Powered by Cognero Page 2

3 Write each fraction as an expression using a negative exponent other than 1. esolutions Manual - Powered by Cognero Page 3

4 Write each decimal using a negative exponent. 30. The minimum thickness of Saturn s A ring is onetenth kilometer. esolutions Manual - Powered by Cognero Page 4

5 31. The diameter of a typical atom is centimeter. Write the decimal using a negative exponent Evaluate each expression if n 3, p 2, and q esolutions Manual - Powered by Cognero Page 5

6 The table below shows the average lengths of different objects. a. How many times as long is a virus than an atom? b. About how many viruses would fit across a pinhead? c. A football field is about 10 2 meters long. How many times as long is this than a cell? esolutions Manual - Powered by Cognero Page 6

7 a. To find how many times longer a virus is than an atom, divide the length of a virus by the length of an atom. common substances. So, a virus is 10 3 or 1000 times as long as an atom. b. To find how many viruses would fit across a pinhead, divide the length of a pinhead by the length of a virus. So, 10 4 or 10,000 viruses would fit across a pinhead. c. To find how many times as long a football field is than a cell, divide the length of a football field by the length of a cell. a. Which substance in the table has the greatest hydrogen ion concentration? How many times as great is that hydrogen ion concentration than that of egg whites? b. Which substance has a hydrogen ion concentration of one millionth? c. How many times as great is the hydrogen ion concentration of coffee as the hydrogen ion concentration of pure water? a. Since is the largest value in the hydrogen ion concentration column, coffee has the greatest hydrogen ion concentration. To find how many times as great the hydrogen ion concentration of coffee is than that of egg whites, divide the concentration for coffee by the concentration for egg whites. So, a football field is 10 6 or 1,000,000 times as long as a cell. a or 1000 times b or 10,000 c or 1,000,000 times 42. The ph of a substance is a measure of its acidity. The ph scale ranges from 0 to 14, with a ph of 7 being neutral. As the ph decreases, the substance is more acidic. The table shows the ph of several common substances. So, the hydrogen ion concentration of coffee is 10 3 or 1000 times as great as that of egg whites. b. One million 1,000,000 or 10 6, so one millionth or Milk has a hydrogen ion concentration of 10 6 or one millionth. c. To find how many times as great the hydrogen ion concentration of coffee is than that of pure water, divide the concentration for coffee by the concentration for pure water. esolutions Manual - Powered by Cognero Page 7

8 So, the hydrogen ion concentration of coffee is 10 2 or 100 times as great as that of pure water. a. coffee, 10 3 or 1000 times b. milk c or 100 times 43. Be Precise A grain of sand has a volume of about cubic millimeter. a. Write this number using a negative exponent. b. An empty bottle used to create sand art can hold about grains of sand. What is the approximate volume of the sand art bottle? c. If one cubic centimeter is equal to 10 3 cubic millimeters, how many cubic centimeters of sand will the bottle hold? a. b. To find the approximate volume of the sand art bottle, multiply the number of grains of sand the bottle can hold by the volume of one grain of sand. So, the approximate volume of the sand art bottle is 10 6 mm 3. c. To find how many cubic centimeters of sand the bottle will hold, divide the volume of the bottle by the number of cubic millimeters in one cubic centimeter. 44. The wavelength of X-rays are between 1 and 10 nanometers. If a nanometer is equal to a billionth of a meter, express the greatest wavelength of an X-ray in meters. Write the expression using a negative exponent. 1 nanometer meter To express the greatest wavelength of an X-ray in meters, multiply the greatest wavelength in nanometers by the number of meters in 1 nanometer. The greatest wavelength of an X-ray is 10 8 meters m 45. The shortest period of time ever measured directly was a light burst of a laser lasting about second. Write this decimal as a fraction and as a power of ten. ; The bottle will hold 10 3 or 1000 cm 3 of sand. a b mm 3 c cm Multiple Representations In this problem, you will explore negative exponents when using powers of 10, 10 1 or 0.1. esolutions Manual - Powered by Cognero Page 8

9 a. Table Copy and complete the table shown. b. Reasoning Do you notice a pattern between the negative powers of 10 and their decimal equivalents? Explain. c. Words Write a rule that could be used to find the decimal equivalent of any negative power of 10. d. Numbers Use the rule from part c to find the value of a. b. yes; Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For example, d Model with Mathematics A pizza is cut into 25 equal pieces. Write a fraction that represents one piece of the pizza. Then write this fraction as a power that has a negative exponent. Show the steps you would take to write the fraction as a power. b. There is a pattern. Sample answer: As the exponents decrease, the number of zeros in the decimal places increase. c. Sample answer: The number of zeros in the decimal equivalent is equal to one less than the absolute value of the negative exponent. For example, d should have or 12 zeros in the decimal equivalent a. esolutions Manual - Powered by Cognero Page 9

10 48. Find the Error Mahala is evaluating the expression Find her mistake and correct it. a. Mahala added parentheses to the numbers in the expression ( 2) 2 and ( 2) b. The correct solution is or. Mahala added parentheses to the numbers in the expression. The correct solution is or. 49. Reason Inductively Consider the following sets of numbers: Set 1: 2 2, ( 2) 2, ( 2) 2, 2 2 Set 2: 2 3, ( 2) 3, ( 2) 3, 2 3 a. Simplify each expression in Set 1. Which expressions, if any, are equal? b. Simplify each expression in Set 2. Which expressions, if any, are equal? c. Explain why the number of equal expressions is different for each list. d. Finish the conjecture: 2 x ( 2) x, if and only if. e. Finish the conjecture: ( 2) x 2 x, if and only if. a. None of the expressions are equal. c. Sample answer: When you square either a positive or a negative value, the answer is positive. When you cube a positive value, you get a positive and when you cube a negative value, you get a negative. d. 2 x ( 2) x, if and only if x is an even number. e. ( 2) x 2 x, if and only if x is an even number. a.,,, 2 2 4; 2 2 ( 2) 2 and ( 2) b.,,, ; None of the expressions are equal. c. Sample answer: When you square either a positive or a negative value, the answer is positive. When you cube a positive value, you get a positive and when you cube a negative value, you get a negative. d. x is an even number e. x is an even number esolutions Manual - Powered by Cognero Page 10

11 50. Persevere with Problems Compare and contrast x n and x n where x 0. Then give a numerical example to show the relationship. They are multiplicative inverses. Sample answer: 2 4 and 2 4 are multiplicative inverses because 2 4, and. They are multiplicative inverses. Sample answer: 2 4 and 2 4 are multiplicative inverses because 2 4, and. 51. Justify Conclusions Investigate the fraction. Does it increase or decrease as the value of n increases? Explain. Sample answer: If n 3, If n 4, 52. Building on the Essential Question Explain the difference between the expressions ( 3) 4 and 3 4. Sample answer: ( 3) 4 is the same as ( 3)( 3)( 3) ( 3) or is the same as or. Sample answer: ( 3) 4 is the same as ( 3)( 3)( 3) ( 3) or is the same as or. 53. DNA contains the genetic code of an organism. The length of a DNA strand is about 10 7 meter. Which of the following represents the length of the DNA strand as a decimal? A m B m C m D m So, as the value of n increases, the value of decreases. Sample answer: If n 3, If n 4, The length of the DNA strand is meters. Choice C is correct. C So, as the value of n increases, the value of decreases. esolutions Manual - Powered by Cognero Page 11

12 54. When simplified, 2 5 is equal to which of the following? F 32 G 56. SHORT RESPONSE It takes light seconds to travel one mile. Write as a fraction and as a power of 10. H J 32 ; 10 6 Choice H is correct. H 55. Which of the following shows the expressions 4 0, 4 2, 4 2, and 4 3 in order from least to greatest? A 4 3, 4 2, 4 2, 4 0 B 4 0, 4 2, 4 3, 4 2 C 4 2, 4 0, 4 2, 4 3 D 4 3, 4 2, 4 0, 4 2 Since the bases are all equal, arrange the exponents from least to greatest. 3 < 2 < 0 < 2, so 4 3 < 4 2 < 4 0 < 4 2 Choice D is correct. 57. The table shows the elevations of geographic places in relation to sea level. Order the elevations from least to greatest. Write the elevations in the order they would appear from left to right on a number line. So, the elevations ordered from least to greatest are 509, 505, 435, , 505, 435, 410 D esolutions Manual - Powered by Cognero Page 12

13 58. Find each sum or difference. Write in simplest form esolutions Manual - Powered by Cognero Page 13

14 Financial Literacy Rami withdrew $75 from his savings each month for 3 months. What is his total withdrawal for the 3 months? The integer 75 represents a withdrawal of $75. To find the total withdrawal for 3 months, multiply 75 by A total withdrawal of $225 or $225. a withdrawal of $225 or $225 Evaluate each expression n if n p if p xy if x 4 and y 5 60 esolutions Manual - Powered by Cognero Page 14

15 68. 8st if s 2 and t 6 96 Write each expression using exponents The base 15 is a factor 5 times. So, the exponent is (p + 2)(p + 2)(p + 2) The base (p + 2) is a factor 3 times. So, the exponent is 3. (p + 2)(p + 2)(p + 2) (p + 2) 3 (p + 2) 3 Find each product bc bc bc bc bc bc bc bc The base bc is a factor 8 times. So, the exponent is 8. bc bc bc bc bc bc bc bc (bc) 8 or b 8 c (bc) 8 or b 8 c x x y y y y x 2 y esolutions Manual - Powered by Cognero Page 15

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4-2 Negative Exponents

4-2 Negative Exponents 1. 2. 3. Write each expression using a positive exponent. 8. 9. When a baseball is hit, it comes in contact with the bat for less than 0.001 of a second. Write 0.001 using a negative exponent other than