Roots and Powers. Written by: Bette Kreuz Edited by: Science Learning Center Staff
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1 Roots and Powers Written by: Bette Kreuz Edited by: Science Learning Center Staff
2 The objectives for this module are to: 1. Raise exponential numbers to a power. 2. Extract the root of an exponential number. a) Extract the root when the exponent is not evenly divisible by the root. b) Extract the root when the power has a numerator 1.
3 Introduction Frequently, in your science courses, you will be asked to solve problems that involve raising a number to a power (such as squaring or cubing) or extracting a root (such as the square root or cube root) in order to obtain a numerical answer. Also, in dealing with extremely large or small numbers, it is helpful to work with the number in exponential form. For these reasons, it is often necessary to know how to raise exponential numbers to a power & how to extract the root of a number in exponential form.
4 PART 1 RAISING EXPONENTIAL NUMBERS TO A POWER When a number, say, 200, is squared, it is written as The number 2, slightly above and to the right of the number 200, is a Power. From your previous math courses you know that: = 200 x 200 = 40,000
5 PART 1 RAISING EXPONENTIAL NUMBERS TO A POWER In general, when a number is raised to a power, it means that the number is multiplied by itself the number of times given by the power. For example, two hundred cubed, or raised to a power of three, is written: The numerical value of is: 200 x 200 x 200 = 8,000,000
6 PART 1 RAISING EXPONENTIAL NUMBERS TO A POWER Three additional examples are: (0.20) 2 = 0.20 x 0.20 = (10) 4 =10 x 10 x 10 x 10 = 10,000 (0.20) 3 =0.20 x 0.20 x 0.20 = (Note that there are two significant figures in the answer when there are two significant figures in the number being raised to a power.)
7 PART 1 RAISING EXPONENTIAL NUMBERS TO A POWER An exponential number, for example 2.00 x 10 2, can also be raised to a power. The terms "power" and "exponent" can be used interchangeably for the superscript on 10. In this module, we will use the term exponent when referring to the superscript on 10 and power for the superscript on an exponential number: (2.00 x 10 2 ) 2 Power Exponent
8 Study the examples below. Each example is written in several different, yet equivalent, exponential forms: (2.00 x 10 2 ) 2 = (2.00 x 10 2 ) x (2.00 x 10 2 ) = 4.00 x 10 4 (2.00 x 10 2 ) 3 = (2.00 x 10 2 ) x (2.00 x 10 2 ) x (2.00 x 10 2 ) = 8.00 x 10 6 (2.00 x 10-1 ) 2 = (2.00 x 10-1 ) x (2.00 x 10-1 ) = 4.00 x 10-2 (2.00 x 10-1 ) 3 = (2.00 x 10-1 ) x (2.00 x 10-1 ) x (2.00 x 10-1 ) = 8.00 x 10-3
9 From studying the preceding examples you can see that there are two rules for raising an exponential to a power. These rules are: 1. (the value of the exponent of 10 in the answer) = (the exponent of 10 in the original number) x (the value of the power) (2.00 x 10 2 ) 2 =4.00 x x 2 = 4 (2.00 x 10 2 ) 3 =8.00 x x 3 = 6 (2.00 x 10-1 ) 2 =4.00 x x 2 = -2 (2.00 x 10-1 ) -3 =0.125 x x -3 = 3 *Note that it is necessary to carry through the appropriate sign into the answer.
10 2. (the value of the coefficient in the answer) = (the coefficient of the original exponential number) multiplied by itself the number of times indicated by the power (2.00 x 10 2 ) 2 = 4.00 x x 2.00 = 4.00 (2.00 x 10 2 ) 3 = 8.00 x x 2.00 x 2.00 = 8.00 (2.0 x 10-1 ) 2 =4.0 x x 2.0 = 4.0 (2.00 x 10-1 ) -3 = x 10 3 hint: (2.00 )= (. )... =.
11 PART 1 RAISING EXPONENTIAL NUMBERS TO A POWER The general formula for raising an exponential number to a power is shown below: (10 A ) B = 10 A x B OR (Z x 10 A ) B = (Z) B x 10 A x B RULE 1 RULE 2
12 PART 1 RAISING EXPONENTIAL NUMBERS TO A POWER Below are two worked examples of exponential numbers raised to a power. (4.10 x 10-2 ) 3 = (4.10) 3 x 10-2x3 = 68.9 x 10-6 OR 6.89 x 10-5 (6.00 x 10 4 ) 2 = (6.00) 2 x 10 4x2 = 36.0 x 10 8 OR 3.60 x 10 9
13 PROBLEM SET I Now it's your turn. On a separate piece of paper work the problems below to obtain the correct numerical answer. You may check your work with the answers at the end of the module. 1) (3.00 x 10-3 ) 4 2) (5.6 x 10 8 ) 2 3) (1.00 x 10-4 ) 6
14 PART 2a - EXTRACTING THE ROOT OF AN EXPONENTIAL NUMBER To obtain a square root of a number, say, 400 ( 400 ), we try to find the number that when multiplied by itself two times produces the original number (20 x 20 = 400). This process is just the opposite of raising a number to a power.
15 PART 2a - EXTRACTING THE ROOT OF AN EXPONENTIAL NUMBER Look at the examples below: (Remember that 4 1/2 = 4 0.5, 16 1/4 = , etc.) 400 = 20 OR (400) 0.5 = 20 OR (400) 1/2 = 20 SINCE 20 X 20 = 400 8,000,000 = 200 OR (8,000,000) 1/3 = 200 SINCE 200 X 200 X 200 = 8,000,000
16 The operation of root extraction can be designated either with a radical sign where n indicates the root ( ) or with a fractional power,, where n again represents the root. 64 = 64 1/2 = = 8 125= 125 1/3 = = 5 81 = 81 1/4 = = 3 Usually, in science courses, the fractional power designation is used and this designation will be used in the rest of this module.
17 FOR EXAMPLE: 10 = (10 3 ) 1/3 = 10 1 = = (10 12 ) 1/3 = 10 4 = 10,000 (10-16 ) 0.25 = 10 = (10-16 ) 1/4 = 10-4 = Study the two examples on the next slide & you will notice that the same rules used to raise exponential numbers to a power can be used to extract a root from exponential numbers.
18 EXAMPLE I: (4.00 x 10 6 ) 0.5 = (4.00 x 10 6 ) 1/2 =(4.00) 1/2 x 10 (6 x ½) = (4.00) 1/2 x 10 6/2 = (4.00) 1/2 x 10 3 = 2.00 x 10 3 EXAMPLE II: (8.00 x 10-9 ) 1/3 = (8.00) 1/3 x 10-9 x 1/3 = (8.00) 1/3 x 10-9/3 = (8.00) 1/3 x 10-3 = 2.00 x 10-3
19 Rule 1. Multiply the fraction designating the root by the power of ten to obtain the power of ten in the final answer: EXAMPLE I 6 x 1/2 = 3 EXAMPLE II -9 x 1/3 = -3 Rule 2. Take the appropriate root of the coefficient: EXAMPLE I (4.00) 1/2 = 2.00 EXAMPLE II (8.00) 1/3 = 2.00
20 Extracting roots poses a special problem when the exponent is not evenly divisible by the root given in the fraction. For example, (9.00 x 10 5 ) 0.5 = (9.00 x 10 5 ) 1/2 = (9.00) 1/2 x 10 (5 x ½) = 3.00 x 10 5/2 OR 3.00 x This answer (10 5/2 or ) is not in standard exponential form and is an awkward expression to work with. In such cases, it is necessary to change the coefficient to a power of 10 that is evenly divisible by the root we are extracting as is shown in the next slide.
21 9.00 x 10 5 = 90.0 x 10 4 THEN: (9.00 x 10 5 ) 1/2 = (90.0 x 10 4 ) 1/2 = (90.0) 1/2 x 10 4x½ = (90.0) 1/2 x 10 4/2 = (90.0) 1/2 x 10 2 = 9.49 x 10 2 Before we demonstrate additional examples, we briefly review (next 3 slides) the rules about moving the decimal point in the coefficient & changing the power of 10 without changing the value of the number
22 These rules are: 1) If the decimal is moved to the right, subtract from the exponent of 10 the number equal to the number of places the decimal was moved. (1.60 x 10 5 ) 1/2 = (16.0 x ) 1/2 = (16.0 x 10 4 ) 1/2 (4.700 x 10 7 ) 1/4 = (4700 x ) 1/4 = (4700 x 10 4 ) 1/4
23 2) If the decimal is moved to the left, add to the exponent of 10 the number equal to the number of places the decimal was moved. (3.42 x 10 5 ) 1/3 = (0.342 x ) 1/3 = (0.342 x 10 6 ) 1/3 (1.00 x 10 8 ) 1/3 = (0.100 x ) 1/3 = (0.100 x 10 9 ) 1/3
24 Note that it really doesn't matter which way the decimal point is moved. You will get the same answer either way. For example: OR = (2500 x 10-4 ) 1/2 DECIMAL MOVED TO RIGHT (250.0 x 10-3 ) 1/2 =.5000 x 10 0 = (25.00 x 10-2 ) 1/2 DECIMAL MOVED TO LEFT (3789 x 10 0 ) 1/3 [ Note: (10 0 ) 1/3 = 10 0 = 1] = ( x 10-3 ) 1/3 DECIMAL MOVED TO THE RIGHT = (3.789 x 10 3 ) 1/3 DECIMAL MOVED TO THE LEFT = x 10 1
25 PART 2a EXTRACTING THE ROOT OF AN EXPONENTIAL NUMBER The general formula for extracting the root of an exponential number is shown below: (Z x 10 A ) 1/N RULE 1 (Z) 1/N x 10 A x 1/N (Z) 1/N x 10 A/N RULE 2 Make sure that the exponent of 10 is evenly divisible by N. If it isn t, change the exponent by moving the decimal.
26 PROBLEM SET II It s your turn again. Work the problems and check your work with the answers at the end of the module. 1) (3.60 x 10 5 ) 1/2 OR (3.60 x 10 5 ) 0.5 2) (2.70 x 10 7 ) 1/3 3) (1.00 x ) -1/4 OR (1.00 x ) ) (1.60 x 10 5 ) 1/4 OR (1.60 x 10 5 ) 0.25
27 PART 2b EXTRACTING THE ROOT OF AN EXPONENTIAL NUMBER What do you do when you have powers where the numerator is not 1? FOR EXAMPLE: (9.0 x 10 4 ) 1.5 OR (9.0 x 10 4 ) 3/2 = (9.0 x 10 4 ) 3 x 1/2 = ( )
28 You can multiply the number in parentheses by itself the number of times that the numerator indicates, and then take the root that the denominator indicates as you did using the rules from the previous section. (9.000 x 10 4 ) 3/2 = (9.00 x 10 4 ) 3 x 1/2 = [(9.00 x 10 4 ) x (9.00 x 10 4 ) x (9.00 x 10 4 )] 1/2 =(729 x ) 1/2 = (7.290 x ) 1/2 = x 10 7
29 To help the manipulation of the number and the exponent to get an "easy" number to work with, consider the following 2 ideas: 1. Try extracting the root first & then carry out the power term indicated in the problem. For example: (4.0 x 10 0 ) 3/2 = [(4.0 x 10 0 ) 1/2 ] 3 EXTRACT THE SQUARE ROOT = [2.0 x 10 0 ] 3 RAISE THE ANSWER TO THE 3 rd POWER = 8.0 x 10 0
30 2. In extracting a root consider moving the decimal point to get an "easy" root OR changing the exponent to get a number easily divisible by the root being taken. (Do whichever makes the math less complicated!) Example 1: = (64 x 10 2 ) 1/2 ( Easy root) (6.4 x 10 3 ) 1/2 8.0 x 10 1 = (0.64 x 10 4 ) 1/2 (The square root is less obvious here)
31 Example 2: (3.43 x 10 4 ) 1/3 = (34.3 x 10 3 ) 1/3 = ( x 10 6 ) 1/3 Both give an exponent evenly divisible by 3, BUT, the root of one may be easier to estimate than the other. (34.3 x 10 3 ) 1/3 ( x 10 6 ) 1/3 The root is between 3 3 (27) and 4 3 (64) The root is between (0.027) and (0.064) = 0.32 x 10 2 = 3.2 x 10 1 Both answers are the same. = 3.2 x 10 1
32 PROBLEM SET III Using the Easy Root/Make the Math Less Complicated approach, try the following problems. Check your answers at the end of the module. 1) (8.00 x 10 0 ) 4/3 2) (3.0 x 10-1 ) 4/2
33 CALCULATOR SUPPLEMENT: If you are using a calculator with only a square root button on it and run across a problem like the following: (1.728 x 10 9 ) 1/4 OR (1.728 x 10 9 ).25 It may be helpful to think of 1/4 as (1/2 x 1/2) (1.728 x 10 9 ) 1/2 x 1/2 OR ((1.728 x 10 9 ) 1/2 ) 1/2 and just take the square root of the number in parentheses twice! (once for each time it is multiplied by the 1/2 power.) ((17.28 x 10 8 ) 1/2 ) 1/2 = (4.157 x 10 4 ) 1/2 = x 10 2 Notice that this will only work for numbers where the power has a denominator that is a power of 2, such as -1/4, -1/2, 0, 1/2, 1/4, 1/8, 1/16, 1/32, etc.
34 PROBLEM SET IV If you the Calculator Supplement was useful to you because your calculator only has a square root button, practice doing the following problems on your calculator: 1) (10-8 ) -1/4 OR (10-8 ) ) (409.6 x 10 1 ) 1/4 OR (409.6 x 10 1 ).25 3) (656.1 x 10 1 ) 1/8 4) (2.401 x ) 1/2 OR (2.401 x ) 0.5 5) (8.55 x 10 0 ) 3/4 OR (8.55 x 10 0 ) 0.75
35 PROBLEM SET I ANSWERS 1) 81.0 x OR 8.10 x ) 31 x OR 3.1 x ) 1.00 x 10-24
36 PROBLEM SET II ANSWERS 1) (36.0 x 10 4 ) 1/2 = 6.00 x ) (27.0 x 10 6 ) 1/3 = 3.00 x ) 1.00 x ) (16.0 x 10 4 ) 1/4 = 2.00 x 10 1
37 PROBLEM SET III ANSWERS 1) 16.0 x ) 9.0 x 10-2
38 PROBLEM SET IV ANSWERS 1) 1.0 x ) x ) x ) x ) 5.00 x 10 0
39 Done! If you have any questions or are having difficulty solving these problems, ask the assistant in the Science Learning Center for help. If you feel competent in raising exponential numbers to a power and extracting roots from exponential numbers, ask the assistant for the post-test.
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