Mathematics 10 — Exercise Skilled practice · Page 6 Q1

Full question

Simplify the following expression.

Step-by-step solution

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

   $$\frac{2\times i+1}{3-i}$$

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

   $$i+3$$

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

   $$\left(\frac{2\times i+1}{3-i}\right)\times \left(\frac{i+3}{i+3}\right)$$

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

   $$\frac{\left(i+3\right)\times \left(2\times i+1\right)}{\left(3-i\right)\times \left(i+3\right)}$$

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.

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

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

   $$1\times 3+1\times i+2\times 3\times i+2\times 1\times ii$$

8. Real × real: multiply the numerical coefficients.

   $$1\times 3=3$$

9. Real × imaginary: multiply the numerical coefficients.

   $$1\times i=i$$

10. Imaginary × real: multiply the numerical coefficients.

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

11. Imaginary × imaginary: multiply the numerical coefficients.

    $$2\times 1\times ii=2\times i^2$$

12. Substitute these four products into the expansion.

    $$3+i+6\times i+2\times i^2$$

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

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

14. Multiply its coefficient by −1.

    $$2\times \left(-1\right)=-2$$

15. Rewrite the expansion after this sign calculation.

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

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

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

17. Calculate the sum of the real terms.

    $$3-2=1$$

18. Calculate the sum of the imaginary coefficients.

    $$1+6=7$$

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

    $$7\times i+1$$

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.

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

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

    $$3\times 3+3\times 1\times i-3\times i-1\times ii$$

23. Real × real: multiply the numerical coefficients.

    $$3\times 3=9$$

24. Real × imaginary: multiply the numerical coefficients.

    $$3\times 1\times i=3\times i$$

25. Imaginary × real: multiply the numerical coefficients.

    $$-3\times i=-3\times i$$

26. Imaginary × imaginary: multiply the numerical coefficients.

    $$-1\times ii=-i^2$$

27. Substitute these four products into the expansion.

    $$9+3\times i-3\times i-i^2$$

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

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

29. Multiply its coefficient by −1.

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

30. Rewrite the expansion after this sign calculation.

    $$9+3\times i-3\times i+1$$

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

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

32. Calculate the sum of the real terms.

    $$9+1=10$$

33. Calculate the sum of the imaginary coefficients.

    $$3-3=0$$

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

    $$0\times i+10$$

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

    $$10=10$$

36. Put the calculated numerator above the calculated denominator.

    $$\frac{7\times i+1}{10}$$

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

    $$\frac{1}{10}+\frac{7\times i}{10}$$

38. Move i outside its numerical fraction.

    $$\frac{1}{10}+\left(\frac{7}{10}\right)\times i$$

39. Simplify the real coefficient fraction.

    $$\frac{1}{10}=\frac{1}{10}$$

40. Simplify the imaginary coefficient fraction.

    $$\frac{7}{10}=\frac{7}{10}$$

41. Combine the reduced real and imaginary terms.

    $$\frac{7\times i}{10}+\frac{1}{10}$$

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

    $$\frac{7\times i\times \left(i+2\right)}{10}+\frac{i+2}{10}$$

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

    $$\frac{1}{10}\times 2+\frac{i}{10}+\frac{14\times i}{10}+\frac{7\times i^2}{10}$$

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

    $$\frac{1\times 1}{1\times 10}$$

45. Calculate these two products separately.

    $$\frac{1}{10}$$

46. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{10}=\frac{1}{10}$$

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

    $$\frac{1\times 2}{10\times 1}$$

48. Calculate these two products separately.

    $$\frac{2}{10}$$

49. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=2$$

50. Show the two divisions explicitly.

    $$\frac{\frac{2}{2}}{\frac{10}{2}}$$

51. Evaluate the numerator division.

    $$\frac{2}{2}=1$$

52. Evaluate the denominator division.

    $$\frac{10}{2}=5$$

53. Write the reduced fraction.

    $$\frac{1}{5}$$

54. Real × real: multiply the numerical coefficients.

    $$\frac{1}{10}\times 2=\frac{1}{5}$$

55. Real × imaginary: multiply the numerical coefficients.

    $$\frac{i}{10}=\frac{i}{10}$$

56. Imaginary × real: multiply the numerical coefficients.

    $$\frac{14\times i}{10}=\frac{7\times i}{5}$$

57. Imaginary × imaginary: multiply the numerical coefficients.

    $$\frac{7\times i^2}{10}=\frac{7\times i^2}{10}$$

58. Substitute these four products into the expansion.

    $$\frac{1}{5}+\frac{i}{10}+\frac{7\times i}{5}+\frac{7\times i^2}{10}$$

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

    $$\frac{7\times i^2}{10}=\frac{7}{10}\times \left(-1\right)$$

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

    $$\frac{7\times 1}{1\times 10}$$

61. Calculate these two products separately.

    $$\frac{7}{10}$$

62. Calculate this operation before combining the remaining terms.

    $$7\times \frac{1}{10}=\frac{7}{10}$$

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

    $$\frac{7\times \left(-1\right)}{10\times 1}$$

64. Calculate these two products separately.

    $$-\frac{7}{10}$$

65. Multiply its coefficient by −1.

    $$\frac{7}{10}\times \left(-1\right)=-\frac{7}{10}$$

66. Rewrite the expansion after this sign calculation.

    $$\frac{1}{5}+\frac{i}{10}+\frac{7\times i}{5}-\frac{7}{10}$$

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

    $$\frac{1}{5}-\frac{7}{10}+i\times \left(\frac{1}{10}+\frac{7}{5}\right)$$

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

    $$\frac{1\times 1}{1\times 5}$$

69. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{5}=\frac{1}{5}$$

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

    $$\frac{\left(-1\right)\times 7}{1\times 10}$$

71. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{7}{10}=-\frac{7}{10}$$

72. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=10$$

73. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1}{5}=\frac{1\times 2}{5\times 2}$$

74. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{7}{10}=\frac{\left(-7\right)\times 1}{10\times 1}$$

75. Keep the common denominator and add the signed numerators.

    $$\frac{2-7}{10}=-\frac{5}{10}$$

76. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

77. Show the two divisions explicitly.

    $$\frac{-\frac{5}{5}}{\frac{10}{5}}$$

78. Evaluate the numerator division.

    $$-\frac{5}{5}=-1$$

79. Evaluate the denominator division.

    $$\frac{10}{5}=2$$

80. Write the reduced fraction.

    $$-\frac{1}{2}$$

81. Calculate the sum of the real terms.

    $$\frac{1}{5}-\frac{7}{10}=-\frac{1}{2}$$

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

    $$\frac{7\times 1}{1\times 5}$$

83. Calculate these two products separately.

    $$\frac{7}{5}$$

84. Calculate this operation before combining the remaining terms.

    $$7\times \frac{1}{5}=\frac{7}{5}$$

85. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1}{10}=\frac{1\times 1}{10\times 1}$$

86. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{7}{5}=\frac{7\times 2}{5\times 2}$$

87. Calculate the converted numerator; the denominator is now common.

    $$\frac{14}{10}$$

88. Keep the common denominator and add the signed numerators.

    $$\frac{1+14}{10}=\frac{15}{10}$$

89. Show the two divisions explicitly.

    $$\frac{\frac{15}{5}}{\frac{10}{5}}$$

90. Evaluate the numerator division.

    $$\frac{15}{5}=3$$

91. Write the reduced fraction.

    $$\frac{3}{2}$$

92. Calculate the sum of the imaginary coefficients.

    $$\frac{1}{10}+\frac{7}{5}=\frac{3}{2}$$

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

    $$\frac{3\times i}{2}-\frac{1}{2}$$

Answer

The final answer is shown below.