Mathematics 10 — Exercise Worked examples · Example 6

Full question

Evaluate the expression.

Step-by-step solution

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

   $$\frac{4\times i+3}{5-7\times i}$$

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

   $$7\times i+5$$

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

   $$\left(\frac{4\times i+3}{5-7\times i}\right)\times \left(\frac{7\times i+5}{7\times i+5}\right)$$

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

   $$\frac{\left(4\times i+3\right)\times \left(7\times i+5\right)}{\left(5-7\times i\right)\times \left(7\times i+5\right)}$$

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.

   $$4\times i\times \left(7\times i+5\right)+3\times \left(7\times i+5\right)$$

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

   $$3\times 5+3\times 7\times i+4\times 5\times i+4\times 7\times ii$$

8. Real × real: multiply the numerical coefficients.

   $$3\times 5=15$$

9. Real × imaginary: multiply the numerical coefficients.

   $$3\times 7\times i=21\times i$$

10. Imaginary × real: multiply the numerical coefficients.

    $$4\times 5\times i=20\times i$$

11. Imaginary × imaginary: multiply the numerical coefficients.

    $$4\times 7\times ii=28\times i^2$$

12. Substitute these four products into the expansion.

    $$15+21\times i+20\times i+28\times i^2$$

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

    $$28\times i^2=28\times \left(-1\right)$$

14. Multiply its coefficient by −1.

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

15. Rewrite the expansion after this sign calculation.

    $$15+21\times i+20\times i-28$$

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

    $$15-28+i\times \left(21+20\right)$$

17. Calculate the sum of the real terms.

    $$15-28=-13$$

18. Calculate the sum of the imaginary coefficients.

    $$21+20=41$$

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

    $$41\times i-13$$

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.

    $$-7\times i\times \left(7\times i+5\right)+5\times \left(7\times i+5\right)$$

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

    $$5\times 5+5\times 7\times i-7\times 5\times i-7\times 7\times ii$$

23. Real × real: multiply the numerical coefficients.

    $$5\times 5=25$$

24. Real × imaginary: multiply the numerical coefficients.

    $$5\times 7\times i=35\times i$$

25. Imaginary × real: multiply the numerical coefficients.

    $$-7\times 5\times i=-35\times i$$

26. Imaginary × imaginary: multiply the numerical coefficients.

    $$-7\times 7\times ii=-49\times i^2$$

27. Substitute these four products into the expansion.

    $$25+35\times i-35\times i-49\times i^2$$

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

    $$-49\times i^2=\left(-49\right)\times \left(-1\right)$$

29. Multiply its coefficient by −1.

    $$\left(-49\right)\times \left(-1\right)=49$$

30. Rewrite the expansion after this sign calculation.

    $$25+35\times i-35\times i+49$$

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

    $$25+49+i\times \left(35-35\right)$$

32. Calculate the sum of the real terms.

    $$25+49=74$$

33. Calculate the sum of the imaginary coefficients.

    $$35-35=0$$

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

    $$0\times i+74$$

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

    $$74=74$$

36. Put the calculated numerator above the calculated denominator.

    $$\frac{41\times i-13}{74}$$

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

    $$-\frac{13}{74}+\frac{41\times i}{74}$$

38. Move i outside its numerical fraction.

    $$-\frac{13}{74}+\left(\frac{41}{74}\right)\times i$$

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

    $$\frac{\left(-13\right)\times 1}{1\times 74}$$

40. Calculate these two products separately.

    $$-\frac{13}{74}$$

41. Calculate this operation before combining the remaining terms.

    $$\left(-13\right)\times \frac{1}{74}=-\frac{13}{74}$$

42. Simplify the real coefficient fraction.

    $$-\frac{13}{74}=-\frac{13}{74}$$

43. Simplify the imaginary coefficient fraction.

    $$\frac{41}{74}=\frac{41}{74}$$

44. Combine the reduced real and imaginary terms.

    $$\frac{41\times i}{74}-\frac{13}{74}$$

Answer

The final answer is shown below.