9-4 Negative Exponents

Transcription

1 Write each expression using a positive exponent BASEBALL 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 Write each fraction as an expression using a negative exponent other than 1. Evaluate each expression if x 4 and y esolutions Manual - Powered by Cognero Page 1

2 Write each expression using a positive exponent esolutions Manual - Powered by Cognero Page 2

3 22. Write each fraction as an expression using a negative exponent other than Write each decimal using a negative exponent. 30. SPACE The minimum thickness of Saturn s A ring is one tenth kilometer SCIENCE The diameter of a typical atom is centimeter Evaluate each expression if n 3, p 2, and q esolutions Manual - Powered by Cognero Page 3

4 esolutions Manual - Powered by Cognero Page 4

5 SCIENCE 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? 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. 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. So, a football field is 10 6 or 1,000,000 times as long as a cell. esolutions Manual - Powered by Cognero Page 5

6 42. SCIENCE 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. 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. Compare the hydrogen ion concentration of milk, ph 6, to that of coffee which has a ph PHYSICAL SCIENCE 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. As the ph increases by 1, the concentration of hydrogen ions is multiplied by or divided by 10. d. 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. 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. As the ph increases by 1, describe what happens to the concentration of hydrogen ions. d. 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, the hydrogen ion concentration of coffee is 10 2 or 100 times as great as that of pure water. esolutions Manual - Powered by Cognero Page 6

7 44. SAND A grain of sand has a volume of about cubic millimeters. 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. 45. MULTIPLE REPRESENTATIONS In this problem, you will explore negative exponents. When using powers of 10, 10 1 or 0.1. a. Tabular Copy and complete the table shown. b. Verbal Do you notice a pattern between the negative powers of 10 and their decimal equivalents? Explain. c. Verbal Write a verbal rule that could be used to find the decimal equivalent of any negative power of 10. d. Numerical Use the rule from part c to find the value of a. The bottle will hold 10 3 or 1000 cm 3 of sand. 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 esolutions Manual - Powered by Cognero Page 7

8 46. SCIENCE 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 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. 50. The greatest wavelength of an X-ray is 10 8 meters. Use the Product of Powers and Quotient of Powers rules to simplify each expression OPEN ENDED Write a power that has a negative exponent and show the steps you take to write the power as a fraction. 48. Sample answer: 52. ERROR ANALYSIS Jeannette and Mahala are evaluating the expression Is either of them correct? Explain your reasoning. Mahala did not follow the order of operations. She multiplied the whole numbers first. She should have first performed the operation with the exponent and then multiplied. Jeannette has the correct solution. 53. REASONING Consider the following sets of esolutions Manual - Powered by Cognero Page 8

9 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. 54. CHALLENGE 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. 55. REASONING Investigate the fraction. Does it increase or decrease as the value of n increases? Explain. Sample answer: If n 3, If n 4, So, as the value of n increases, the value of decreases ( 2) 2 and ( 2) b. 56. WRITING IN MATH 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. 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. esolutions Manual - Powered by Cognero Page 9

10 57. 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 60. SHORT RESPONSE It takes light seconds to travel one mile. Write as a fraction and as a power of 10. The length of the DNA strand is meters. Choice C is correct. 58. When simplified, 2 5 is equal to which of the following? F 32 G H J 32 Choice H is correct. 59. 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. 61. ARTS AND CRAFTS When a piece of paper is cut in half, the result is two smaller pieces of paper. When the two smaller pieces are stacked and then cut, the result is four pieces of paper. The number of resulting sheets of paper after c cuts is 2 c. a. How many more pieces of paper are there if a piece of paper is cut and stacked 8 times than when a piece of paper is cut and stacked 5 times? b. A stack of 500 sheets of notebook paper is about 1 inch thick. How thick would your stack be if you were able to make 10 cuts? a. After 8 cuts, there are 2 8 or 256 sheets. After 5 cuts, there are 2 5 or 32 sheets. So, there are or 224 more pieces of paper if 8 cuts are made than when 5 cuts are made. b. Cut Number The stack would be 1024 inches or 85 feet after 10 cuts. Sheets of Paper , , , , , , Thickness (inches) esolutions Manual - Powered by Cognero Page 10

11 ALGEBRA Factor each monomial FOOD The results of a survey about favorite hamburger condiments are shown in the table below. Which condiment was chosen by the most people? Explain fg State the slope and y-intercept of each equation x + y 3 The slope is 2 and the y-intercept is 3. Convert each part to a percent, then compare. Mustard: 22% 22% Ketchup: 40% Relish: % Since 40% > 22% > 20%, the group that said they prefer ketchup is largest. Find each product x + 4y The slope is and the y-intercept is y 4 The slope is 0 and the y-intercept is 4. esolutions Manual - Powered by Cognero Page 11

12 , 000 esolutions Manual - Powered by Cognero Page 12