Mathematics 10 — Exercise Review · Q5 (iii)

Full question

For the given z₁,z₂, verify the displayed conjugate property.

Step-by-step solution

1. Start with the original quotient. For an inverse, the numerator is 1.

   $$\frac{4\times i+3}{3\times i+2}$$

2. Find the conjugate of the denominator: keep its real part and reverse the imaginary sign.

   $$2-3\times i$$

3. Multiply by the conjugate over itself; this fraction equals 1.

   $$\left(\frac{2-3\times i}{2-3\times i}\right)\times \left(\frac{4\times i+3}{3\times i+2}\right)$$

4. Multiply numerators together and denominators together. Keep their brackets intact.

   $$\frac{\left(2-3\times i\right)\times \left(4\times i+3\right)}{\left(2-3\times i\right)\times \left(3\times i+2\right)}$$

5. Calculate the numerator first.

6. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

   $$4\times i\times \left(2-3\times i\right)+3\times \left(2-3\times i\right)$$

7. Distribute each of those products. Keep all four multiplications visible.

   $$3\times 2+3\times \left(-1\right)\times 3\times i+4\times 2\times i+4\times \left(-3\right)\times ii$$

8. Real × real: multiply the numerical coefficients.

   $$3\times 2=6$$

9. Real × imaginary: multiply the numerical coefficients.

   $$3\times \left(-1\right)\times 3\times i=-9\times i$$

10. Imaginary × real: multiply the numerical coefficients.

    $$4\times 2\times i=8\times i$$

11. Imaginary × imaginary: multiply the numerical coefficients.

    $$4\times \left(-3\right)\times ii=-12\times i^2$$

12. Substitute these four products into the expansion.

    $$6-9\times i+8\times i-12\times i^2$$

13. In the last term replace i² by −1.

    $$-12\times i^2=\left(-12\right)\times \left(-1\right)$$

14. Multiply its coefficient by −1.

    $$\left(-12\right)\times \left(-1\right)=12$$

15. Rewrite the expansion after this sign calculation.

    $$6-9\times i+8\times i+12$$

16. Group the real terms and factor i out of the imaginary terms.

    $$6+12+i\times \left(-9+8\right)$$

17. Calculate the sum of the real terms.

    $$6+12=18$$

18. Calculate the sum of the imaginary coefficients.

    $$-9+8=-1$$

19. Insert both calculated sums in a + bi form.

    $$18-i$$

20. Now calculate the denominator independently.

21. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$3\times i\times \left(2-3\times i\right)+2\times \left(2-3\times i\right)$$

22. Distribute each of those products. Keep all four multiplications visible.

    $$2\times 2+2\times \left(-1\right)\times 3\times i+3\times 2\times i+3\times \left(-3\right)\times ii$$

23. Real × real: multiply the numerical coefficients.

    $$2\times 2=4$$

24. Real × imaginary: multiply the numerical coefficients.

    $$2\times \left(-1\right)\times 3\times i=-6\times i$$

25. Imaginary × real: multiply the numerical coefficients.

    $$3\times 2\times i=6\times i$$

26. Imaginary × imaginary: multiply the numerical coefficients.

    $$3\times \left(-3\right)\times ii=-9\times i^2$$

27. Substitute these four products into the expansion.

    $$4-6\times i+6\times i-9\times i^2$$

28. In the last term replace i² by −1.

    $$-9\times i^2=\left(-9\right)\times \left(-1\right)$$

29. Multiply its coefficient by −1.

    $$\left(-9\right)\times \left(-1\right)=9$$

30. Rewrite the expansion after this sign calculation.

    $$4-6\times i+6\times i+9$$

31. Group the real terms and factor i out of the imaginary terms.

    $$4+9+i\times \left(-6+6\right)$$

32. Calculate the sum of the real terms.

    $$4+9=13$$

33. Calculate the sum of the imaginary coefficients.

    $$-6+6=0$$

34. Insert both calculated sums in a + bi form.

    $$0\times i+13$$

35. The imaginary coefficient in the denominator is zero, so the denominator is real.

    $$13=13$$

36. Put the calculated numerator above the calculated denominator.

    $$\frac{18-i}{13}$$

37. Separate the real term and imaginary term into two fractions with the same denominator.

    $$\frac{18}{13}+\frac{-i}{13}$$

38. Move i outside its numerical fraction.

    $$\frac{18}{13}+\left(-\frac{1}{13}\right)\times i$$

39. Simplify the real coefficient fraction.

    $$\frac{18}{13}=\frac{18}{13}$$

40. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 1}{1\times 13}$$

41. Calculate these two products separately.

    $$-\frac{1}{13}$$

42. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{13}=-\frac{1}{13}$$

43. Simplify the imaginary coefficient fraction.

    $$-\frac{1}{13}=-\frac{1}{13}$$

44. Combine the reduced real and imaginary terms.

    $$\frac{18}{13}-\frac{i}{13}$$

45. Take the conjugate of that result by changing the sign of its imaginary coefficient.

    $$\frac{i}{13}+\frac{18}{13}$$

46. Now combine the conjugates of the original numbers independently.

47. Start with the original quotient. For an inverse, the numerator is 1.

    $$\frac{3-4\times i}{2-3\times i}$$

48. Find the conjugate of the denominator: keep its real part and reverse the imaginary sign.

    $$3\times i+2$$

49. Multiply by the conjugate over itself; this fraction equals 1.

    $$\left(\frac{3-4\times i}{2-3\times i}\right)\times \left(\frac{3\times i+2}{3\times i+2}\right)$$

50. Multiply numerators together and denominators together. Keep their brackets intact.

    $$\frac{\left(3-4\times i\right)\times \left(3\times i+2\right)}{\left(2-3\times i\right)\times \left(3\times i+2\right)}$$

51. Calculate the numerator first.

52. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$-4\times i\times \left(3\times i+2\right)+3\times \left(3\times i+2\right)$$

53. Distribute each of those products. Keep all four multiplications visible.

    $$3\times 2+3\times 3\times i-4\times 2\times i-4\times 3\times ii$$

54. Real × real: multiply the numerical coefficients.

    $$3\times 2=6$$

55. Real × imaginary: multiply the numerical coefficients.

    $$3\times 3\times i=9\times i$$

56. Imaginary × real: multiply the numerical coefficients.

    $$-4\times 2\times i=-8\times i$$

57. Imaginary × imaginary: multiply the numerical coefficients.

    $$-4\times 3\times ii=-12\times i^2$$

58. Substitute these four products into the expansion.

    $$6+9\times i-8\times i-12\times i^2$$

59. In the last term replace i² by −1.

    $$-12\times i^2=\left(-12\right)\times \left(-1\right)$$

60. Multiply its coefficient by −1.

    $$\left(-12\right)\times \left(-1\right)=12$$

61. Rewrite the expansion after this sign calculation.

    $$6+9\times i-8\times i+12$$

62. Group the real terms and factor i out of the imaginary terms.

    $$6+12+i\times \left(9-8\right)$$

63. Calculate the sum of the real terms.

    $$6+12=18$$

64. Calculate the sum of the imaginary coefficients.

    $$9-8=1$$

65. Insert both calculated sums in a + bi form.

    $$i+18$$

66. Now calculate the denominator independently.

67. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$-3\times i\times \left(3\times i+2\right)+2\times \left(3\times i+2\right)$$

68. Distribute each of those products. Keep all four multiplications visible.

    $$2\times 2+2\times 3\times i-3\times 2\times i-3\times 3\times ii$$

69. Real × real: multiply the numerical coefficients.

    $$2\times 2=4$$

70. Real × imaginary: multiply the numerical coefficients.

    $$2\times 3\times i=6\times i$$

71. Imaginary × real: multiply the numerical coefficients.

    $$-3\times 2\times i=-6\times i$$

72. Imaginary × imaginary: multiply the numerical coefficients.

    $$-3\times 3\times ii=-9\times i^2$$

73. Substitute these four products into the expansion.

    $$4+6\times i-6\times i-9\times i^2$$

74. In the last term replace i² by −1.

    $$-9\times i^2=\left(-9\right)\times \left(-1\right)$$

75. Multiply its coefficient by −1.

    $$\left(-9\right)\times \left(-1\right)=9$$

76. Rewrite the expansion after this sign calculation.

    $$4+6\times i-6\times i+9$$

77. Group the real terms and factor i out of the imaginary terms.

    $$4+9+i\times \left(6-6\right)$$

78. Calculate the sum of the real terms.

    $$4+9=13$$

79. Calculate the sum of the imaginary coefficients.

    $$6-6=0$$

80. Insert both calculated sums in a + bi form.

    $$0\times i+13$$

81. The imaginary coefficient in the denominator is zero, so the denominator is real.

    $$13=13$$

82. Put the calculated numerator above the calculated denominator.

    $$\frac{i+18}{13}$$

83. Separate the real term and imaginary term into two fractions with the same denominator.

    $$\frac{18}{13}+\frac{i}{13}$$

84. Move i outside its numerical fraction.

    $$\frac{18}{13}+\left(\frac{1}{13}\right)\times i$$

85. Simplify the real coefficient fraction.

    $$\frac{18}{13}=\frac{18}{13}$$

86. Simplify the imaginary coefficient fraction.

    $$\frac{1}{13}=\frac{1}{13}$$

87. Combine the reduced real and imaginary terms.

    $$\frac{i}{13}+\frac{18}{13}$$

88. Both results are equal.

    $$\frac{i}{13}+\frac{18}{13}=\frac{i}{13}+\frac{18}{13}$$

Answer

The final answer is shown below.