Chapter 1. Foundations of Algebra 1.1. Signed Numbers, Exponents and Order of Operations KYOTE Standards: CR 1, CR 2; CA 1 Exponents Multiplication can be viewed as repeated addition. For example, 3⋅ 4 can be viewed as the sum of 3 4's or the sum of 4 3's. Thus 3⋅ 4 = 4 + 4 + 4 =12 or 3⋅ 4 = 3+ 3+3+3 =12 Exponents can be used to express repeated multiplication. For example, the symbol 34 is used to represent the product of 4 3's. Thus 34 = 3⋅3⋅3⋅3 = 81 The symbol 34 is read “ 3 to the fourth power” or simply “ 3 to the fourth.” The number 3 is called the base and the number 4 is called the exponent . We note that the base and exponent cannot in general be interchanged without changing the exponential expression being computed. For example, 34 = 81 is not equal to 43 = 4⋅ 4⋅ 4 = 64. Example 1. Calculate (−2)3 ⋅ 72 Solution . Using the definition of exponential expressions, we have (−2)3 ⋅ 72 = (−2)⋅(−2)⋅(−2)⋅ 7⋅ 7 = (−8)⋅ 49 = −392 Order of Operations The order in which operations are performed to calculate the value of an arithmetic expression is critically important. Let’s look at a few examples to see why. What is the value of 7− 3⋅ 5? Is it (7 −3)⋅ 5 = 4⋅ 5 = 20 or is it 7− (3⋅ 5) = 7−15 = −8? What is the value of 5− 9 − 2? Is it (5− 9)− 2 = −4 − 2 = −6 or is it 5− (9 − 2) = 5− 7 = −2? What is the value of 5⋅ 23? Is it (5⋅ 2)3 =103 =1000 or is it 5⋅(23 ) = 5⋅8 = 40? What is the value of 1+ 24 ? Is it (1+ 2)4 = 34 = 81 or is it 1+(24 ) =1+16 =17? 8 It’s clear that we need some agreement about the order in which mathematical expressions should be evaluated. This agreement is stated below. Order of Operations Rule To evaluate mathematical expressions, carry out the operations in the following order: 1. Perform operations within parentheses. 2. Perform all exponentiation operations. 3. Perform all multiplications and additions from left to right. 4. Perform all additions and subtractions from left to right. A few examples should illustrate this rule. We strongly recommended that you do these calculations without a calculator to develop fluency with signed numbers and exponents as well as order of operations. You are then welcome to check your answer using a calculator. You will notice that all calculators are programmed to follow the order of operations rule. Example 2. Simplify (a) −3− 7+ 2 (b) −3−(7+ 2) (c) 36 ÷ 2 ⋅3 (d) 36 ÷ (2 ⋅3) Solution . (a) Since −3− 7+ 2 involves only additions and subtractions, we perform all operations from left to right to obtain −3− 7+ 2 = (−3− 7)+ 2 = −10 + 2 = −8 (b) Since −3−(7+ 2) involves parentheses, we perform the operations within parentheses first to obtain −3− (7+ 2) = −3− 9 = −12 (c) Since 36 ÷ 2 ⋅3 involves only multiplications and divisions, we perform all operations from left to right to obtain 36 ÷ 2⋅3 = (36 ÷ 2)⋅3 =18⋅3 = 54 (d) Since 36 ÷ (2 ⋅3) involves parentheses, we perform the operations within parentheses first to obtain 36 ÷ (2⋅3) = 36 ÷ 6 = 6 9 Example 3. Simplify (a) −6 − 3⋅ 23 +(−4)2 (b) (−6 − 3)⋅ 23 + (−4)2 Solution . (a) We perform all exponentiations first, then all multiplications, and finally all additions and subtractions from left to right to obtain −6 − 3⋅ 23 + (−4)2 = −6 − 3⋅8+16 Calculate 23 , (−4)2 = −6 − 24 +16 Calculate 3⋅8 = −30 +16 Calculate −6 − 24 = −14 Calculate −30 +16 (b) We first calculate −6 − 3 in parentheses. We next perform all exponentiations, then all multiplications, and then additions and subtractions to obtain (−6 − 3)⋅ 23 + (−4)2 = (−9)⋅ 23 + (−4)2 Calculate −6 − 3 = −9⋅8+16 Calculate 23 , (−4)2 = −72 +16 Calculate −9 ⋅8 = −56 Calculate −72 +16 Example 4. Find the value of x3 − 2x2 y − y when x = −1 and y = −3. Solution . We substitute x = −1 and y = −3 into x3 − 2x2 y − y and follow the order of operations rule to obtain (−1)3 − 2(−1)2 (−3)− (−3) Substitute x = −1 and y = −3 Into expression x3 − 2x2 y − y = −1− 2(1)(−3)+ 3 Calculate (−1)3 , (−1)2 = −1+ 6 + 3 Calculate (−2)⋅1⋅(−3) = 8 Calculate −1+ 6 +3 Exercise Set 1.1 Simplify without using a calculator. 1. 3− 4 − 5 2. 3−(4 − 5) 3. 7+ 2⋅8 4. (7+ 2)⋅8 5. −2(6 − 9)−1 6. −2⋅ 6 − 9 −1 10 7. −9(2 − 5)− 7 8. −9 ⋅ 2 − (5− 7) 9. 30 ÷ 2⋅ 5 10. 30 ÷ (2 ⋅5) 11. (30 ÷ 2)⋅ 5 12. 6 −8÷ 2 13. (6 −8)÷ 2 14. 6 −(8÷ 2) 15. 48÷8÷ 2 16. 48÷ (8÷ 2) 17. (48÷8)÷ 2 18. 3⋅ 5− 3⋅ 7 19. 3(5− 7) 20. 3⋅(5− 3)⋅ 7 Simplify without using a calculator. 21. 34 22. 53 23. 25 24. 72 25. (−4)2 26. −42 27. (−2)3 ⋅(−3)2 28. −23 ⋅(−3)2 29. (2 − 7)2 30. 22 − 72 31. (3− 5)3 32. 33 − 53 33. −7+3⋅ 22 34. −7+(3⋅ 2)2 35. (−7+ 3)⋅ 22 36. 8− 2⋅32 37. (8− 2)⋅32 38. 8−(2⋅3)2 39. 20 ÷ 22 40. (20 ÷ 2)2 41. 3⋅ 52 42. (3⋅ 5)2 43. 32 ⋅ 52 44. (3+ 5)2 45. 32 + 52 46. −6 + 5⋅(−3)2 11 47. (−6 + 5)⋅(−3)2 48. −6 − 5⋅32 Find the value of the algebraic expression at the specified values of its variable or variables. 49. x2 − 2x − 5; x = −3 50. 4 − x ; x = −3 51. x − 2(1− 3x); x = −1 52. x3 − 4x2 − 5x; x = −1 53. (x − y)2; x =1, y = −3 54. x − (y − x2 ); x = −3, y = 2 55 . −x + 3y2; x = −6, y = −2 56. 3+ xy2; x = −5, y = −2 57. (3+ x)y2; x = −5, y = −2 58. 3+(xy)2; x = −5, y = −2 12 1.2. Prime Numbers, GCF and LCM KYOTE Standards: Foundational Content A positive integer can be factored in a several ways. For example, 12 can be factored in three ways as 12 =1⋅12, 12 = 2 ⋅6 and 12 = 3⋅ 4 . The factors of 12 are the six positive integers 1, 2,3, 4,6,12 . Positive integers like 12 with more than two factors are called composite numbers. On the other hand, 7 can be factored in only one way as 7 =1⋅ 7. The factors of 7 are the two positive integers 1, 7 . Positive integers like 7 with exactly two factors are called prime numbers. The formal definitions are given below. Definition 1. A prime number is a positive integer that has exactly two factors, itself and 1. A composite number is a positive integer that has more than two factors. Example 1. Find the number of factors of the given positive integer and determine whether that integer is prime or composite. (a) 36 (b) 17 (c) 1 (d) 65 Solution . (a) Note that 36 =1⋅36, 36 = 2 ⋅18, 36 = 3⋅12, 36 = 4⋅ 9 , 36 = 6⋅ 6. Thus 36 has nine factors 1, 2,3, 4, 6, 9,12,18,36 and is a composite number. (b) Note that 17 =1⋅17but no other factorizations. Thus 17 has two factors 1,17 and is a prime number. (c) Note that 1=1⋅1 but no other factorizations. Thus 1has only one factor and is therefore not a prime number. (d) Note that 65 =1⋅ 65, 65 = 5⋅13. Thus 65 has four factors and is a composite number. The following result is an important building block in both arithmetic and algebra. Fundamental Theorem of Arithmetic . Any positive integer greater than 1 can be written uniquely as the product of powers of prime numbers. This product of powers of prime numbers is called the prime factorization of the positive integer. Let’s consider a few examples. It is easy to check that the positive integer 12 can be written 12 = 22 ⋅3 and it is easy to find this factorization since 12 is a relatively small integer. It is easy to check that 924 = 22 ⋅3⋅ 7⋅11 with a calculator, but it is more difficult to find its prime factorization. Example 2 shows how to obtain the prime factorization of a relatively small positive integer by inspection and simple calculations. Example 3 shows how to obtain the prime factorization of a relatively large positive integer in a more deliberate fashion. 13 Example 2. Find the prime factorization of the given positive integer. (a) 72 (b) 23 (c) 180 Solution . (a) We know that 72 = 8⋅ 9 and factorization of 8 = 23 and 9 = 32 gives the prime factorization 72 = 23 ⋅32 . (b) A search for factors of 23 yields only 23 and 1. Thus 23 is a prime number and its prime factorization is simply 23 itself. (c) We see immediately that 10 is a factor of 180 and we write 180 =18⋅10 . We note that 18 = 2⋅ 9 = 2 ⋅32 and 10 = 2 ⋅ 5 to obtain the prime factorization of 180 = 22 ⋅32 ⋅ 5. Example 3. Find the prime factorization of 6552 . Solution. The prime factorization of 6552 is more difficult to obtain but illustrates a general method of finding prime factorizations.