Mathematics 10 — Exercise 1.3 · Q2 (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.

   $$\frac{7\times i+2}{4-3\times i}$$

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

   $$3\times i+4$$

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

   $$\left(\frac{7\times i+2}{4-3\times i}\right)\times \left(\frac{3\times i+4}{3\times i+4}\right)$$

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

   $$\frac{\left(3\times i+4\right)\times \left(7\times i+2\right)}{\left(4-3\times i\right)\times \left(3\times i+4\right)}$$

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.

   $$7\times i\times \left(3\times i+4\right)+2\times \left(3\times i+4\right)$$

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

   $$2\times 4+2\times 3\times i+7\times 4\times i+7\times 3\times ii$$

8. Real × real: multiply the numerical coefficients.

   $$2\times 4=8$$

9. Real × imaginary: multiply the numerical coefficients.

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

10. Imaginary × real: multiply the numerical coefficients.

    $$7\times 4\times i=28\times i$$

11. Imaginary × imaginary: multiply the numerical coefficients.

    $$7\times 3\times ii=21\times i^2$$

12. Substitute these four products into the expansion.

    $$8+6\times i+28\times i+21\times i^2$$

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

    $$21\times i^2=21\times \left(-1\right)$$

14. Multiply its coefficient by −1.

    $$21\times \left(-1\right)=-21$$

15. Rewrite the expansion after this sign calculation.

    $$8+6\times i+28\times i-21$$

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

    $$8-21+i\times \left(6+28\right)$$

17. Calculate the sum of the real terms.

    $$8-21=-13$$

18. Calculate the sum of the imaginary coefficients.

    $$6+28=34$$

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

    $$34\times i-13$$

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.

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

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

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

23. Real × real: multiply the numerical coefficients.

    $$4\times 4=16$$

24. Real × imaginary: multiply the numerical coefficients.

    $$4\times 3\times i=12\times i$$

25. Imaginary × real: multiply the numerical coefficients.

    $$-3\times 4\times i=-12\times i$$

26. Imaginary × imaginary: multiply the numerical coefficients.

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

27. Substitute these four products into the expansion.

    $$16+12\times i-12\times i-9\times i^2$$

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.

    $$16+12\times i-12\times i+9$$

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

    $$16+9+i\times \left(12-12\right)$$

32. Calculate the sum of the real terms.

    $$16+9=25$$

33. Calculate the sum of the imaginary coefficients.

    $$12-12=0$$

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

    $$0\times i+25$$

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

    $$25=25$$

36. Put the calculated numerator above the calculated denominator.

    $$\frac{34\times i-13}{25}$$

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

    $$-\frac{13}{25}+\frac{34\times i}{25}$$

38. Move i outside its numerical fraction.

    $$-\frac{13}{25}+\left(\frac{34}{25}\right)\times i$$

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

    $$\frac{\left(-13\right)\times 1}{1\times 25}$$

40. Calculate these two products separately.

    $$-\frac{13}{25}$$

41. Calculate this operation before combining the remaining terms.

    $$\left(-13\right)\times \frac{1}{25}=-\frac{13}{25}$$

42. Simplify the real coefficient fraction.

    $$-\frac{13}{25}=-\frac{13}{25}$$

43. Simplify the imaginary coefficient fraction.

    $$\frac{34}{25}=\frac{34}{25}$$

44. Combine the reduced real and imaginary terms.

    $$\frac{34\times i}{25}-\frac{13}{25}$$

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

    $$-\frac{34\times i}{25}-\frac{13}{25}$$

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

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

    $$\frac{2-7\times i}{3\times i+4}$$

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

    $$4-3\times i$$

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

    $$\left(\frac{2-7\times i}{3\times i+4}\right)\times \left(\frac{4-3\times i}{4-3\times i}\right)$$

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

    $$\frac{\left(2-7\times i\right)\times \left(4-3\times i\right)}{\left(4-3\times i\right)\times \left(3\times i+4\right)}$$

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.

    $$-7\times i\times \left(4-3\times i\right)+2\times \left(4-3\times i\right)$$

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

    $$2\times 4+2\times \left(-1\right)\times 3\times i-7\times 4\times i-7\times \left(-3\right)\times ii$$

54. Real × real: multiply the numerical coefficients.

    $$2\times 4=8$$

55. Real × imaginary: multiply the numerical coefficients.

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

56. Imaginary × real: multiply the numerical coefficients.

    $$-7\times 4\times i=-28\times i$$

57. Imaginary × imaginary: multiply the numerical coefficients.

    $$-7\times \left(-3\right)\times ii=21\times i^2$$

58. Substitute these four products into the expansion.

    $$8-6\times i-28\times i+21\times i^2$$

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

    $$21\times i^2=21\times \left(-1\right)$$

60. Multiply its coefficient by −1.

    $$21\times \left(-1\right)=-21$$

61. Rewrite the expansion after this sign calculation.

    $$8-6\times i-28\times i-21$$

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

    $$8-21+i\times \left(-6-28\right)$$

63. Calculate the sum of the real terms.

    $$8-21=-13$$

64. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 28=-28$$

65. Calculate the sum of the imaginary coefficients.

    $$-6-28=-34$$

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

    $$-34\times i-13$$

67. Now calculate the denominator independently.

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

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

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

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

70. Real × real: multiply the numerical coefficients.

    $$4\times 4=16$$

71. Real × imaginary: multiply the numerical coefficients.

    $$4\times \left(-1\right)\times 3\times i=-12\times i$$

72. Imaginary × real: multiply the numerical coefficients.

    $$3\times 4\times i=12\times i$$

73. Imaginary × imaginary: multiply the numerical coefficients.

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

74. Substitute these four products into the expansion.

    $$16-12\times i+12\times i-9\times i^2$$

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

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

76. Multiply its coefficient by −1.

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

77. Rewrite the expansion after this sign calculation.

    $$16-12\times i+12\times i+9$$

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

    $$16+9+i\times \left(-12+12\right)$$

79. Calculate the sum of the real terms.

    $$16+9=25$$

80. Calculate the sum of the imaginary coefficients.

    $$-12+12=0$$

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

    $$0\times i+25$$

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

    $$25=25$$

83. Put the calculated numerator above the calculated denominator.

    $$\frac{-34\times i-13}{25}$$

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

    $$-\frac{13}{25}+\frac{-34\times i}{25}$$

85. Move i outside its numerical fraction.

    $$-\frac{13}{25}+\left(-\frac{34}{25}\right)\times i$$

86. Simplify the real coefficient fraction.

    $$-\frac{13}{25}=-\frac{13}{25}$$

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

    $$\frac{\left(-34\right)\times 1}{1\times 25}$$

88. Calculate these two products separately.

    $$-\frac{34}{25}$$

89. Calculate this operation before combining the remaining terms.

    $$\left(-34\right)\times \frac{1}{25}=-\frac{34}{25}$$

90. Simplify the imaginary coefficient fraction.

    $$-\frac{34}{25}=-\frac{34}{25}$$

91. Combine the reduced real and imaginary terms.

    $$-\frac{34\times i}{25}-\frac{13}{25}$$

92. Both results are equal.

    $$-\frac{34\times i}{25}-\frac{13}{25}=-\frac{34\times i}{25}-\frac{13}{25}$$

Answer

The final answer is shown below.