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1 s3 multiplication and division of signed numbers

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  • 1. Multiplication and Division of Signed Numbers
  • 2. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers
  • 3. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product.
  • 4. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ;
  • 5. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ;
  • 6. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ; Two numbers with the same sign multiplied yield positive products.
  • 7. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ; Two numbers with the same sign multiplied yield a positive product. Two numbers with opposite signs multiplied yield a negative product.
  • 8. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ; Two numbers with the same sign multiplied yield a positive product. Two numbers with opposite signs multiplied yield a negative product. Example A. a. 5 * (4) = –5 * (–4)
  • 9. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ; Two numbers with the same sign multiplied yield a positive product. Two numbers with opposite signs multiplied yield a negative product. Example A. a. 5 * (4) = –5 * (–4) = 20
  • 10. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ; Two numbers with the same sign multiplied yield a positive product. Two numbers with opposite signs multiplied yield a negative product. Example A. a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4)
  • 11. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ; Two numbers with the same sign multiplied yield a positive product. Two numbers with opposite signs multiplied yield a negative product. Example A. a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4) = –20
  • 12. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ; Two numbers with the same sign multiplied yield a positive product. Two numbers with opposite signs multiplied yield a negative product. Example A. a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4) = –20 In algebra, multiplication operation are not always written down explicitly.
  • 13. Multiplication and Division of Signed Numbers Rule for Multiplication of Signed Numbers To multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ; Two numbers with the same sign multiplied yield a positive product. Two numbers with opposite signs multiplied yield a negative product. Example A. a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4) = –20 In algebra, multiplication operation are not always written down explicitly. Instead we use the following rules to identify multiplication operations.
  • 14. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication.
  • 15. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y.
  • 16. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication.
  • 17. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
  • 18. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x. ● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication.
  • 19. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x. ● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication. Hence (x + y)(a + b) = (x + y) * (a + b)
  • 20. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x. ● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication. Hence (x + y)(a + b) = (x + y) * (a + b) However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine.
  • 21. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x. ● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication. Hence (x + y)(a + b) = (x + y) * (a + b) However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15,
  • 22. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x. ● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication. Hence (x + y)(a + b) = (x + y) * (a + b) However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
  • 23. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x. ● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication. Hence (x + y)(a + b) = (x + y) * (a + b) However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8, and –5(–5) = (–5)(–5) = 25,
  • 24. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x. ● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication. Hence (x + y)(a + b) = (x + y) * (a + b) However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8, and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.
  • 25. Multiplication and Division of Signed Numbers ● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y. ● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x. ● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication. Hence (x + y)(a + b) = (x + y) * (a + b) However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8, and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10. To multiply many signed numbers together, we always determine the sign of the product first, then multiply just the numbers themselves. The sign of the product is determined by the following Even–Odd Rules.
  • 26. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive.
  • 27. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative.
  • 28. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative. Example B. a. –1(–2 ) 2 (–1)
  • 29. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative. Example B. a. –1(–2 ) 2 (–1) three negative numbers, so the product is negative
  • 30. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative. Example B. a. –1(–2 ) 2 (–1) = – 4 4 came from 1*2*2*1 (just the numbers) three negative numbers, so the product is negative
  • 31. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative. Example B. a. –1(–2 ) 2 (–1) = – 4 three negative numbers, so the product is negative b. (–2)4
  • 32. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative. Example B. a. –1(–2 ) 2 (–1) = – 4 three negative numbers, so the product is negative b. (–2)4 = (–2 )(–2)(–2)(–2)
  • 33. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative. Example B. a. –1(–2 ) 2 (–1) = – 4 three negative numbers, so the product is negative b. (–2)4 = (–2 )(–2)(–2)(–2) four negative numbers, so the product is positive
  • 34. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative. Example B. a. –1(–2 ) 2 (–1) = – 4 three negative numbers, so the product is negative b. (–2)4 = (–2 )(–2)(–2)(–2) = 16 four negative numbers, so the product is positive
  • 35. Multiplication and Division of Signed Numbers Even-Odd Rule for the Sign of a Product • If there are even number of negative numbers in the multiplication, the product is positive. • If there are odd number of negative numbers in the multiplication, the product is negative. Example B. a. –1(–2 ) 2 (–1) = – 4 three negative numbers, so the product is negative b. (–2)4 = (–2 )(–2)(–2)(–2) = 16 four negative numbers, so the product is positive Fact: A quantity raised to an even power is always positive i.e. xeven is always positive (except 0).
  • 36. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . a b
  • 37. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . Rule for the Sign of a Quotient a b
  • 38. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . a b Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications.
  • 39. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . a b Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. + + = – – = + + + = – = – –
  • 40. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . a b Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. + + = – – = + + – + = – = – Two numbers with the same sign divided yield a positive quotient.
  • 41. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . a b Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. + + = – – = + + – + = – = – Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient.
  • 42. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient. Example C. a. a b + + = – – = + + + = – = – – 20 4 = –20 –4
  • 43. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient. Example C. a. a b + + = – – = + + + = – = – – 20 4 = –20 –4 = 5
  • 44. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient. Example C. a. b . –20 / 4 = 20 / (–4) a b + + = – – = + + + = – = – – 20 4 = –20 –4 = 5
  • 45. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient. Example C. a. b . –20 / 4 = 20 / (–4) = –5 a b + + = – – = + + + = – = – – 20 4 = –20 –4 = 5
  • 46. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient. Example C. a. b . –20 / 4 = 20 / (–4) = –5 a b + + = – – = + + + = – = – – 20 4 = –20 –4 = 5 c. (–6)2 = –4
  • 47. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient. Example C. a. b . –20 / 4 = 20 / (–4) = –5 a b + + = – – = + + + = – = – – 20 4 = –20 –4 = 5 c. (–6)2 = 36 –4 –4
  • 48. Multiplication and Division of Signed Numbers In algebra, a ÷ b is written as a/b or . Rule for the Sign of a Quotient Division of signed numbers follows the same sign-rules for multiplications. Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient. Example C. a. b . –20 / 4 = 20 / (–4) = –5 a b + + = – – = + + + = – = – – 20 4 = –20 –4 = 5 c. (–6)2 = 36 –4 = –4 –9
  • 49. Multiplication and Division of Signed Numbers The Even–Odd Rule applies to more length * and / operations problems.
  • 50. Multiplication and Division of Signed Numbers The Even–Odd Rule applies to more length * and / operations problems. Example D. Simplify. (– 4)6(–1)(–3) (–2)(–5)12
  • 51. Multiplication and Division of Signed Numbers The Even–Odd Rule applies to more length * and / operations problems. Example D. Simplify. (– 4)6(–1)(–3) (–2)(–5)12 five negative numbers so the product is negative
  • 52. Multiplication and Division of Signed Numbers The Even–Odd Rule applies to more length * and / operations problems. Example D. Simplify. (– 4)6(–1)(–3) (–2)(–5)12 = – five negative numbers so the product is negative
  • 53. Multiplication and Division of Signed Numbers The Even–Odd Rule applies to more length * and / operations problems. Example D. Simplify. (– 4)6(–1)(–3) (–2)(–5)12 = – five negative numbers so the product is negative simplify just the numbers 4(6)(3) 2(5)(12)
  • 54. Multiplication and Division of Signed Numbers The Even–Odd Rule applies to more length * and / operations problems. Example D. Simplify. (– 4)6(–1)(–3) (–2)(–5)12 = – five negative numbers so the product is negative simplify just the numbers 4(6)(3) 2(5)(12) = – 3 5
  • 55. Multiplication and Division of Signed Numbers The Even–Odd Rule applies to more length * and / operations problems. Example D. Simplify. (– 4)6(–1)(–3) (–2)(–5)12 = – five negative numbers so the product is negative simplify just the numbers 4(6)(3) 2(5)(12) = – 3 5 Various form of the Even–Odd Rule extend to algebra and geometry. It’s the basis of many decisions and conclusions in mathematics problems. The following is an example of the two types of graphs there are due to this Even–Odd Rule. (Don’t worry about how they are produced.)
  • 56. Multiplication and Division of Signed Numbers The Even Power Graphs vs. Odd Power Graphs of y = xN
  • 57. Multiplication and Division of Signed Numbers Exercise A. Calculate the following expressions. Make sure that you interpret the operations correctly. 1. 3 – 3 2. 3(–3) 3. (3) – 3 4. (–3) – 3 5. –3(–3) 6. –(–3)(–3) 7. (–3) – (–3) 8. –(–3) – (–3) B.Multiply. Determine the sign first. 9. 2(–3) 10. (–2)(–3) 11. (–1)(–2)(–3) 12. 2(–2)(–3) 13. (–2)(–2)(–2) 14. (–2)(–2)(–2)(–2) 15. (–1)(–2)(–2)(–2)(–2) 16. 2(–1)(3)(–1)(–2) C. Simplify. Determine the sign and cancel first. 17. 12 –3 18. –12 –3 19. –24 –8 21. (2)(–6) –8 20. 24 –12 22. (–18)(–6) –9 23. (–9)(6) (12)(–3) 24. (15)(–4) (–8)(–10) 25. (–12)(–9) (– 27)(15) 26. (–2)(–6)(–1) (2)(–3)(–2) 27. 3(–5)(–4) (–2)(–1)(–2) 28. (–2)(3)(–4)5(–6) (–3)(4)(–5)6(–7)

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