Mathematics 10 — Exercise Skilled practice · Page 13 (ii)

Full question

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

Step-by-step solution

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

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

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

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

3. Real × real: multiply the numerical coefficients.

   $$2\times 4=8$$

4. Real × imaginary: multiply the numerical coefficients.

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

5. Imaginary × real: multiply the numerical coefficients.

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

6. Imaginary × imaginary: multiply the numerical coefficients.

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

7. Substitute these four products into the expansion.

   $$8-2\times i+12\times i-3\times i^2$$

8. In the last term replace i² by −1.

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

9. Multiply its coefficient by −1.

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

10. Rewrite the expansion after this sign calculation.

    $$8-2\times i+12\times i+3$$

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

    $$8+3+i\times \left(-2+12\right)$$

12. Calculate the sum of the real terms.

    $$8+3=11$$

13. Calculate the sum of the imaginary coefficients.

    $$-2+12=10$$

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

    $$10\times i+11$$

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

    $$11-10\times i$$

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

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

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

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

    $$2\times 4+2\times 1\times i-3\times 4\times i-3\times 1\times ii$$

19. Real × real: multiply the numerical coefficients.

    $$2\times 4=8$$

20. Real × imaginary: multiply the numerical coefficients.

    $$2\times 1\times i=2\times i$$

21. Imaginary × real: multiply the numerical coefficients.

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

22. Imaginary × imaginary: multiply the numerical coefficients.

    $$-3\times 1\times ii=-3\times i^2$$

23. Substitute these four products into the expansion.

    $$8+2\times i-12\times i-3\times i^2$$

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

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

25. Multiply its coefficient by −1.

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

26. Rewrite the expansion after this sign calculation.

    $$8+2\times i-12\times i+3$$

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

    $$8+3+i\times \left(2-12\right)$$

28. Calculate the sum of the real terms.

    $$8+3=11$$

29. Calculate the sum of the imaginary coefficients.

    $$2-12=-10$$

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

    $$11-10\times i$$

31. Both results are equal.

    $$11-10\times i=11-10\times i$$

Answer

The final answer is shown below.