Math 8 Notes Unit 3: Exponents and Scientific Notation

Transcription

1 Math 8 Notes Unit : Exponents and Scientific Notation Writing Exponents Exponential form: a number is in exponential form when it is written with a base and an exponent. 5 ; the base is 5 and the exponent is. Exponent: represents how many times the base is used as a factor. 5 ; since is the exponent, use 5 as a factor times factors of 5 Base: is the number to be used as the factor. 5 ; since 5 is the base, use it as the factor is used as the factor Power: a number produced by raising a base to an exponent. 5 = 15, so 5 to the third power (also known as 5 cubed ) is 15. Write in exponential form = 5 7 Identify how many times 5 is used as a factor. In this example it is used 7 times. Write in exponential form. Once again, identify how many times the base is being used as a factor. Write q q q q in exponential form q q q q = 8 q 4 Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 1 of 15

2 Evaluating Exponents Syllabus Objective: (.1) The student will evaluate expressions with exponents. To evaluate exponents have students write the problem in expanded form to help alleviate misconceptions. Evaluate 5 5 = = 4 Always be aware of negative signs and parentheses their location can greatly affect your solution. Evaluate and ; 8 8 (8 8) 64 Evaluate: Step 1: substitute Step : simplify Don t forget to follow order of operations. Syllabus Objective: (.5) The student will evaluate expressions with negative exponents. Pattern development is an effective way to introduce the concept of negative and zero exponents. Consider the following pattern that students should have seen previously. Dividing by each time represents one less factor of. You can quickly see the value of the zero exponent. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page of 15

3 Now you can use the same idea to expand the pattern to calculate negative exponents. Negative Exponents Words Numbers Algebra Any number except 0 with a negative exponent equals its reciprocal with the opposite exponent b n 1 b n, where b 0 Zero Exponent Words Numbers Algebra The zero power of any number except 0 equals 1. Evaluate or Simplify and (4 + ).. Re-write without negative exponents.. Add and evaluate the exponent. 4. Find a common denominator and write equivalent fractions. 5. Subtract. 6. Simplify the fraction by writing it as a mixed number. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page of 15

4 Properties of Exponents Syllabus Objective: (.) The student will apply the properties of exponents. CCSS 8.EE.1-1: Know the properties of integer exponents. The factors of a power can be grouped in different ways. Use this fact with your students to explore the properties of exponents. can be written in the following ways: You can also have students write powers in expanded form to help them discover the properties. MULTIPLYING POWERS WITH THE SAME BASE Words Numbers Algebra To multiply powers with the same base, keep the base and add the exponents. Now let s look at what happens when you divide powers with the same base. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 4 of 15

5 Once the students work with a few examples of this nature, they will be able to come up with the following: DIVIDING POWERS WITH THE SAME BASE Words Numbers Algebra To divide powers with the same base, keep the base and subtract the exponents. To see what happens when you raise a power to a power, use the order of operations. RAISING A POWER TO A POWER Words Numbers Algebra To raise a power to a power, keep the base and multiply the exponents. CCSS 8.EE1-: Apply the properties of integer exponents to generate equivalent numerical expressions. Geckos can easily climb smooth vertical surfaces. It has been discovered by biologists that the reason the feet of a gecko are so sticky are the tiny hairs. Each hair is about micrometers long. (A micrometer is 10 meter. Determine the approximate length of one hair in meters. To find the length of one hair in meters, I will multiply the length of the hair in micrometers by the number of micrometers in one meter. I can convert 100 into exponential form. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 5 of 15

6 ( 6) , The length of one hair is about meter. 10, 000 Comparing and Ordering Real Numbers Syllabus Objective: (.) The student will compare real numbers including powers of whole numbers. Syllabus Objective: (.4) The student will order real numbers including powers of whole numbers. One way to order real numbers is to first express each number as a decimal. Write the decimals in order from least to greatest; then write the corresponding real numbers in the same order. Since irrationals have not been addressed in the curriculum yet, we will limit the ordering to rational numbers. Order the following numbers from least to greatest ,,, = 1 Therefore: 0 9,,, is the correct order, which corresponds to Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 6 of 15

7 Order the rational numbers from least to greatest to determine which student is correct. A) Ricardo B) Mary C) Janice D) Frank Answer is D) Frank. Frank Ricardo Mary Janice Scientific Notation Syllabus Objective: (.6) The student will express numbers using scientific notation. CCSS 8.EE. -1: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities. Scientific Notation: the product of two numbers, the first of which is greater than or equal to one, but less than 10 and the second is a power of 10. It is in the form of where and n is an integer. For example:. Have students convert numbers from scientific notation to standard form by first expanding the number. Write in standard form. Think: move the decimal point to the right 5 spaces. Write in standard form. Write in standard form. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 7 of 15

8 Think: Move the decimal 5 spaces to the left. Write in standard form. To convert a number in standard form to scientific notation, rewrite the number as a product of a power of 10 and a number greater than or equal to 1 but less than 10. Write in scientific notation Think: The decimal needs to move spaces to get a number between 1 and 10. Set up scientific notation. So notation is written in scientific Think: the decimal needs to move left to change 7.09 to , so the exponent will be negative. Write 75,000 in scientific notation. How many places do you need to move the decimal point to get a number between 1 and 10? 5 places Think: the decimal needs to move 5 places to the right to change 7.5 into 75,000, so the exponent is positive. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 8 of 15

9 Ordering Numbers Using Scientific Notation To compare numbers written in scientific notation, first compare the powers of 10, then compare the decimal parts. Order , , 8900 from least to greatest. A. First, write all numbers in scientific notation as necessary B. Order the numbers with different powers of Because 10 10, we know and C. Order the numbers with the same power of 10. Because We know , D. Write the original number in order from least to greatest ; 8900; Operations in Scientific Notation CCSS 8.EE4-1: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. One of the properties of quantities with exponents is that numbers with exponents can be added and subtracted only when they have the same base and same exponent. Since all numbers in scientific notation have the same base (10), we need only worry about the exponents. To be added or subtracted, two numbers in scientific notation must be manipulated so that their bases have the same exponent--this will ensure that corresponding digits in their coefficients have the same place value. Here are the steps to adding or subtracting numbers in scientific notation: 1. Determine the number by which to increase the smaller exponent so it is equal to the larger exponent. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 9 of 15

10 . Increase the smaller exponent by this number and move the decimal point of the number with the smaller exponent to the left the same number of places. (i.e. divide by the appropriate power of 10.). Add or subtract the new coefficients. 4. If the answer is not in scientific notation (i.e. if the coefficient is not between 1 and 10) convert it to scientific notation. ( ) + ( ) =? =. The smaller exponent must be increased by = ( ) + ( ) = ( ) 10 8 = is in scientific notation. Thus, ( ) + ( ) = ( ) ( ) =? 1. ( ) = 5. The smaller exponent must be increased by = ( ) ( ) = ( ) 10 = is in scientific notation. Thus, ( ) ( ) = ( ) + ( ) =? = 6. The smaller exponent must be increased by = ( ) + ( ) = ( ) = is in scientific notation. Thus, ( ) + ( ) = ( ) ( ) =? The smaller exponent must be increased by = ( ) ( ) = ( ) 10-5 = Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 10 of 15

11 = in scientific notation. Thus, = Let s add a step where one of the numbers is in decimal form. 480,100 ( ) =? First rewrite 480,100 in scientific notation: = 1. The smaller exponent must be increased by = ( ) ( ) = ( ) 10 6 = is in scientific notation. Thus, 480,100 ( ) = Quantities with exponents can be multiplied and divided easily if they have the same base. Since all numbers in scientific notation have base 10, we can always multiply them and divide them. Steps for Multiplication in Scientific Notation 1) Regroup (using the commutative and associative properties of multiplication) ) Simplify; apply the products of powers property (add exponents) ) If necessary, rewrite in scientific form. Let s try an step step But what happens if the coefficient is more than 10 when using scientific notation? step step Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 11 of 15

12 While our value is correct, our answer is not written in scientific notation. No problem. We must move the decimal point to the left until the coefficient is between 1 and 10. For each place we move the decimal, the exponent will be raised 1 power of 10. So we rewrite step as.0 10 or 10 in scientific notation. More Examples: in scientific notation in scientific notation What happens when the exponent(s) are negative? We still add the exponents, but remember to use the rules of addition of signed numbers Rules for Division in Scientific Notation 1) Regroup as a product of two fractions ) Simplify; apply the products of powers property (subtract exponents) ) If necessary, rewrite in scientific form Let s try an step step 10 6 Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 1 of 15

13 step1 While the value is correct, it is not written in scientific notation. Since the coefficient is not between 1 and 10, we must move the decimal point to the right until the coefficient is between 1 and 10. For each place we move the decimal, the exponent will be lowered 1 power of ten. So we rewrite step = step What happens when the exponents are negative? in scientific notation. We will still subtract the exponents (applying the rules for subtracting signed numbers) Word Problems Using Scientific Notation Using scientific notation, solve the following problems. 8 The speed of light is 10 meters/second. If the sun is approximately meters from Earth, determine how many seconds it takes light to reach the Earth. Express your answer in both standard notation and scientific notation. d To determine the time, t, where t is time, d is distance, and r is rate. So we have r t seconds 500 seconds At this point, we may want to convert this into minutes: 1 minute seconds 8 minutes 60 seconds Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 1 of 15

14 The Earth is a space ship. It travels about 960,000,000 km a year. How far has it traveled in 4,000,000,000 years? To determine the distance, d rt. d 960, 000, 000 4, 000, 000, 000 This computation leaves lots of room for error! However, using scientific notation: d CCSS 8.EE-: Express how many times as much one number expressed in scientific notation is than another km the Earth has traveled At times Neptune is about,700,000,000 miles from the Earth. Mercury is about 15,000,000 miles from Earth. Determine how many times further it is from the Earth to Neptune than it is from Earth to Mercury. 9 Neptune is.7 10 miles from Earth Mercury is miles from Earth Neptune Mercury or 0 times; Neptune is about 0 times as far from Earth as Mercury. CCSS 8.EE4-: Using scientific notation and choose units of appropriate size for measurements of very large or very small quantities. With very large or very small numbers it is important to know what units we are dealing with. 7 For example, having dollars would be more than ten times the amount of money needed 7 to retire while most financial experts would say that cents would only be about half of what is necessary for retirement. As another example, consider the fact that Mars is on average 7 about km from Earth. It would not make sense to report this distance in millimeters because of how huge that number really is. When deciding on which unit of measurement to use in any given situation, just use common sense. If we re talking about the distance between towns, miles will work better than inches. If we re talking about how quickly your toenails grow, meters per day would (hopefully) be inappropriate while millimeters per day might do nicely. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 14 of 15

15 The point is, people strongly prefer numbers that aren't really big or really small. So this is why we have so many different kinds of units! Obviously, our calculators do not give you appropriate units. You need to keep track of what unit of measurement goes with each calculation. Practice choosing units of appropriate size. Some suggestions are: Choose the most appropriate unit of measurement for the given situation. 1. The amount a rock deteriorates: grams per second, grams per minute, or grams per year. The amount of lava coming from a volcano: fluid ounces per hour, cups per hour, or gallons per hour. The speed human hair grows: inches per year, feet per year, or yards per year 4. The growth of coral: centimeters per year, kilometers per day, or kilometers per year 5. The growth of a tree: inches per hour, inches per year, yards per year 6. Speed of a swimming dolphin: centimeters per hour, meters per hour, kilometers per hour 7. The rate of water flow from a shower head: fluid ounces per minute, cups per minute, gallons per minute 8. A cell phone measures kilometers in thickness. Would this be best expressed using: kilometers, meters, or centimeters? 9. The average pace for a biker is centimeters per hour. Would this be best expressed using: kilometers, meters, or centimeters? 10. A bullet travels millimeters per second. Would this be best expressed using: millimeters, centimeters, or meters per second? Scientific Notation on the Calculator CCSS 8.EE4-: Interpret scientific notation that has been generated by technology. In working with calculators, it is important that students recognize scientific notation. Students 4 should recognize that the output of.45e+ is and.5e-4 is Students should know that to enter scientific notation they will use E or EE (scientific notation), * (multiplication), and ^ (exponent) symbols. Math 8, Unit : Exponents Holt: Chapter 4, Sections 1-4 Page 15 of 15