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OpenStax-CNX module: m34871 1 Exponents, Roots, and Factorization of Whole Numbers: Exponents and Roots ∗ Wade Ellis Denny Burzynski This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 † Abstract This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses exponents and roots. By the end of the module students should be able to understand and be able to read exponential notation, understand the concept of root and be able to read root notation, and use a calculator having the yx key to determine a root. 1 Section Overview • • • • • Exponential Notation Reading Exponential Notation Roots Reading Root Notation Calculators 2 Exponential Notation Exponential Notation We have noted that multiplication is a description of repeated addition. Exponential notation is a description of repeated multiplication. Suppose we have the repeated multiplication 8·8·8·8·8 Exponent The factor 8 is repeated 5 times. Exponential notation uses a superscript for the number of times the factor is repeated. The superscript is placed on the repeated factor, 85 , in this case. The superscript is called an exponent. The Function of an Exponent An exponent records the number ∗ † of identical factors that are repeated in a multiplication. Version 1.2: Aug 18, 2010 8:24 pm +0000 http://creativecommons.org/licenses/by/3.0/ http://cnx.org/content/m34871/1.2/ OpenStax-CNX module: m34871 2 2.1 Sample Set A Write the following multiplication using exponents. Example 1 3 · 3. Since the factor 3 appears 2 times, we record this as 32 Example 2 62 · 62 · 62 · 62 · 62 · 62 · 62 · 62 · 62. Since the factor 62 appears 9 times, we record this as 629 Expand (write without exponents) each number. Example 3 124 . The exponent 4 is recording 4 factors of 12 in a multiplication. Thus, 124 = 12 · 12 · 12 · 12 Example 4 7063 . The exponent 3 is recording 3 factors of 706 in a multiplication. Thus, 7063 = 706 · 706 · 706 2.2 Practice Set A Write the following using exponents. Exercise 1 (Solution on p. 10.) Exercise 2 (Solution on p. 10.) Exercise 3 (Solution on p. 10.) 37 · 37 16 · 16 · 16 · 16 · 16 9·9·9·9·9·9·9·9·9·9 Write each number without exponents. Exercise 4 (Solution on p. 10.) Exercise 5 (Solution on p. 10.) 853 47 Exercise 6 1, 7392 (Solution on p. 10.) 3 Reading Exponential Notation In a number such as 85 , Base 8 is called the base. Exponent, Power 5 is called the exponent, or power. 85 is read as "eight to the fth power," or more simply as "eight to the fth," or "the fth power of eight." Squared When a whole number is raised to the second power, it is said to be squared. The number 52 can be read as http://cnx.org/content/m34871/1.2/ OpenStax-CNX module: m34871 3 5 to the second power, or 5 to the second, or 5 squared. Cubed When a whole number is raised to the third power, it is said to be cubed. The number 53 can be read as 5 to the third power, or 5 to the third, or 5 cubed. When a whole number is raised to the power of 4 or higher, we simply say that that number is raised to that particular power. The number 58 can be read as 5 to the eighth power, or just 5 to the eighth. 4 Roots In the English language, the word "root" can mean a source of something. In mathematical terms, the word "root" is used to indicate that one number is the source of another number through repeated multiplication. Square Root We know that 49 = 72 , that is, 49 = 7 · 7. Through repeated multiplication, 7 is the source of 49. Thus, 7 is a root of 49. Since two 7's must be multiplied together to produce 49, the 7 is called the second or square root of 49. Cube Root We know that 8 = 23 , that is, 8 = 2 · 2 · 2. Through repeated multiplication, 2 is the source of 8. Thus, 2 is a root of 8. Since three 2's must be multiplied together to produce 8, 2 is called the third or cube root of 8. We can continue this way to see such roots as fourth roots, fth roots, sixth roots, and so on. 5 Reading Root Notation n There is a symbol used to indicate roots of a number. It is called the radical sign √ The Radical Sign √ n √ n The symbol is called a radical sign and indicates the nth root of a number. We discuss particular roots using the radical sign as follows: Square Root √ number indicates the square root of the number under the radical sign. It is customary to drop the 2 in the radical sign when discussing square roots. The symbol √ is understood to be the square root radical sign.√ 49 = 7 since 7 · 7 = 72 = 49 2 Cube Root √ 3 number indicates the cube root of the number under the radical sign. √ 3 8 = 2 since 2 · 2 · 2 = 23 = 8 Fourth Root √ 4 number indicates the fourth root of the number under the radical sign. √ 4 81 = 3 since 3 · 3 · 3 · 3 = 34 = 81 √ 5 In an expression such as 32 Radical Sign √ is called the radical sign. Index 5 is called the index. (The index describes the indicated root.) Radicand 32 is called the radicand. http://cnx.org/content/m34871/1.2/ OpenStax-CNX module: m34871 4 Radical √ 5 32 is called a radical (or radical expression). 5.1 Sample Set B Find each root. Example 5 √ 25 To determine the square root of 25, we ask, "What whole number squared equals 25?" From our √experience with multiplication, we know this number to be 5. Thus, 25 = 5 2 Check: 5 · 5 = 5 = 25 Example 6 √ 32 To determine the fth root of 32, we ask, "What whole number raised to the fth power equals 32?" This number is 2. √ 5 32 = 2 5 Check: 2 · 2 · 2 · 2 · 2 = 2 = 32 5 5.2 Practice Set B Find the following roots using only a knowledge of multiplication. Exercise 7 √ (Solution on p. 10.) Exercise 8 √ (Solution on p. 10.) Exercise 9 √ (Solution on p. 10.) Exercise 10 √ (Solution on p. 10.) 64 100 3 6 64 64 6 Calculators Calculators with the √ x, y x , and 1/x keys can be used to nd or approximate roots. 6.1 Sample Set C Example 7 Use the calculator to nd √ 121 Display Reads Type 121 Press x √ 121 11 Table 1 Example √ 8 Find 7 2187. http://cnx.org/content/m34871/1.2/ OpenStax-CNX module: m34871 5 Display Reads Type 2187 2187 Press yx 2187 Type 7 7 Press 1/x .14285714 Press = 3 Table 2 √ 7 2187 = 3 (Which means that 37 = 2187 .) 6.2 Practice Set C Use a calculator to nd the following roots. Exercise 11 √ (Solution on p. 10.) Exercise 12 √ (Solution on p. 10.) Exercise 13 √ (Solution on p. 10.) Exercise 14 √ (Solution on p. 10.) 3 4 729 8503056 53361 12 16777216 7 Exercises For the following problems, write the expressions using exponential notation. Exercise 15 (Solution on p. 10.) 4·4 Exercise 16 12 · 12 Exercise 17 (Solution on p. 10.) 9·9·9·9 Exercise 18 10 · 10 · 10 · 10 · 10 · 10 Exercise 19 826 · 826 · 826 (Solution on p. 10.) Exercise 20 3, 021 · 3, 021 · 3, 021 · 3, 021 · 3, 021 Exercise 21 |6 · 6 ·{z· · · · 6} 85 factors of 6 Exercise 22 2 | · 2 ·{z· · · · 2} 112 factors of 2 http://cnx.org/content/m34871/1.2/ (Solution on p. 10.) OpenStax-CNX module: m34871 Exercise 23 6 (Solution on p. 10.) |1 · 1 ·{z· · · · 1} 3,008 factors of 1 For the following problems, expand the terms. (Do not nd the actual value.) Exercise 24 53 Exercise 25 (Solution on p. 10.) 74 Exercise 26 152 Exercise 27 1175 (Solution on p. 10.) Exercise 28 616 Exercise 29 (Solution on p. 10.) 302 For the following problems, determine the value of each of the powers. Use a calculator to check each result. Exercise 30 32 Exercise 31 (Solution on p. 10.) 42 Exercise 32 12 Exercise 33 102 (Solution on p. 10.) Exercise 34 112 Exercise 35 122 (Solution on p. 10.) Exercise 36 132 Exercise 37 152 (Solution on p. 11.) Exercise 38 14 Exercise 39 (Solution on p. 11.) 34 Exercise 40 73 Exercise 41 103 (Solution on p. 11.) Exercise 42 1002 Exercise 43 83 http://cnx.org/content/m34871/1.2/ (Solution on p. 11.) OpenStax-CNX module: m34871 7 Exercise 44 55 Exercise 45 (Solution on p. 11.) 93 Exercise 46 62 Exercise 47 (Solution on p. 11.) 71 Exercise 48 128 Exercise 49 (Solution on p. 11.) 27 Exercise 50 05 Exercise 51 (Solution on p. 11.) 84 Exercise 52 58 Exercise 53 (Solution on p. 11.) 69 Exercise 54 253 Exercise 55 422 (Solution on p. 11.) Exercise 56 313 Exercise 57 155 (Solution on p. 11.) Exercise 58 220 Exercise 59 (Solution on p. 11.) 8162 For the following problems, nd the roots (using your knowledge of multiplication). Use a calculator to check each result. Exercise 60 √ 9 Exercise 61 √ 16 (Solution on p. 11.) Exercise 62 √ 36 Exercise 63 √ 64 Exercise 64 √ 121 http://cnx.org/content/m34871/1.2/ (Solution on p. 11.) OpenStax-CNX module: m34871 8 Exercise 65 √ (Solution on p. 11.) 144 Exercise 66 √ 169 Exercise 67 √ (Solution on p. 11.) 225 Exercise 68 √ 27 3 Exercise 69 √ (Solution on p. 11.) 32 5 Exercise 70 √ 256 4 Exercise 71 √ (Solution on p. 11.) 216 3 Exercise 72 √ 7 1 Exercise 73 √ (Solution on p. 11.) 400 Exercise 74 √ 900 Exercise 75 √ (Solution on p. 11.) 10, 000 Exercise 76 √ 324 Exercise 77 √ 3, 600 For the following problems, use a calculator with the keys (Solution on p. 11.) √ x, y x , and 1/x to nd each of the values. Exercise 78 √ 676 Exercise 79 √ 1, 156 (Solution on p. 11.) Exercise 80 √ 46, 225 Exercise 81 √ 17, 288, 964 (Solution on p. 11.) Exercise 82 √ 3 3, 375 Exercise 83 √ 4 331, 776 (Solution on p. 11.) Exercise 84 √ 8 5, 764, 801 Exercise 85 √ 12 16, 777, 216 Exercise 86 √ 8 16, 777, 216 http://cnx.org/content/m34871/1.2/ (Solution on p. 11.) OpenStax-CNX module: m34871 Exercise 87 √ 10 9, 765, 625 9 (Solution on p. 12.) Exercise 88 √ 4 160, 000 Exercise 89 √ 3 531, 441 (Solution on p. 12.) 7.1 Exercises for Review Exercise 90 () Use the numbers 3, 8, and 9 to illustrate the associative property of addition. Exercise 91 (Solution on p. 12.) () In the multiplication 8 · 4 = 32, specify the name given to the numbers 8 and 4. Exercise 92 () Does the quotient 15 ÷ 0 exist? If so, what is it? Exercise 93 () Does the quotient 0 ÷ 15exist? If so, what is it? Exercise 94 (Solution on p. 12.) () Use the numbers 4 and 7 to illustrate the commutative property of multiplication. http://cnx.org/content/m34871/1.2/ OpenStax-CNX module: m34871 Solutions to Exercises in this Module Solution to Exercise (p. 2) 372 Solution to Exercise (p. 2) 165 Solution to Exercise (p. 2) 910 Solution to Exercise (p. 2) 85 · 85 · 85 Solution to Exercise (p. 2) 4·4·4·4·4·4·4 Solution to Exercise (p. 2) 1, 739 · 1, 739 Solution to Exercise (p. 4) 8 Solution to Exercise (p. 4) 10 Solution to Exercise (p. 4) 4 Solution to Exercise (p. 4) 2 Solution to Exercise (p. 5) 9 Solution to Exercise (p. 5) 54 Solution to Exercise (p. 5) 231 Solution to Exercise (p. 5) 4 Solution to Exercise (p. 5) 42 Solution to Exercise (p. 5) 94 Solution to Exercise (p. 5) 8263 Solution to Exercise (p. 5) 685 Solution to Exercise (p. 6) 13008 Solution to Exercise (p. 6) 7·7·7·7 Solution to Exercise (p. 6) 117 · 117 · 117 · 117 · 117 Solution to Exercise (p. 6) 30 · 30 Solution to Exercise (p. 6) 4 · 4 = 16 Solution to Exercise (p. 6) 10 · 10 = 100 http://cnx.org/content/m34871/1.2/ 10 OpenStax-CNX module: m34871 Solution to Exercise (p. 6) 12 · 12 = 144 Solution to Exercise (p. 6) 15 · 15 = 225 Solution to Exercise (p. 6) 3 · 3 · 3 · 3 = 81 Solution to Exercise (p. 6) 10 · 10 · 10 = 1, 000 Solution to Exercise (p. 6) 8 · 8 · 8 = 512 Solution to Exercise (p. 7) 9 · 9 · 9 = 729 Solution to Exercise (p. 7) 71 = 7 Solution to Exercise (p. 7) 2 · 2 · 2 · 2 · 2 · 2 · 2 = 128 Solution to Exercise (p. 7) 8 · 8 · 8 · 8 = 4, 096 Solution to Exercise (p. 7) 6 · 6 · 6 · 6 · 6 · 6 · 6 · 6 · 6 = 10, 077, 696 Solution to Exercise (p. 7) 42 · 42 = 1, 764 Solution to Exercise (p. 7) 15 · 15 · 15 · 15 · 15 = 759, 375 Solution to Exercise (p. 7) 816 · 816 = 665, 856 Solution to Exercise (p. 7) 4 Solution to Exercise (p. 7) 8 Solution to Exercise (p. 8) 12 Solution to Exercise (p. 8) 15 Solution to Exercise (p. 8) 2 Solution to Exercise (p. 8) 6 Solution to Exercise (p. 8) 20 Solution to Exercise (p. 8) 100 Solution to Exercise (p. 8) 60 Solution to Exercise (p. 8) 34 Solution to Exercise (p. 8) 4,158 Solution to Exercise (p. 8) 24 http://cnx.org/content/m34871/1.2/ 11 OpenStax-CNX module: m34871 Solution to Exercise (p. 8) 4 Solution to Exercise (p. 9) 5 Solution to Exercise (p. 9) 81 Solution to Exercise (p. 9) 8 is the multiplier; 4 is the multiplicand Solution to Exercise (p. 9) Yes; 0 http://cnx.org/content/m34871/1.2/ 12