Mathematics 10 — Exercise Review · Q5 (i)

Full question

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

Step-by-step solution

1. Write both complex numbers with their real and imaginary parts.

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

2. Remove the brackets, preserving the signs of all four terms.

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

3. Place the real terms together. Factor i out of the imaginary terms.

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

4. Calculate the real part only.

   $$3+2=5$$

5. Calculate the imaginary coefficient only.

   $$4+3=7$$

6. Put these two results into a + bi form.

   $$7\times i+5$$

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

   $$5-7\times i$$

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

9. Write both complex numbers with their real and imaginary parts.

   $$-(4\times i-3)-(3\times i-2)$$

10. Remove the brackets, preserving the signs of all four terms.

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

11. Place the real terms together. Factor i out of the imaginary terms.

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

12. Calculate the real part only.

    $$3+2=5$$

13. Calculate this operation before combining the remaining terms.

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

14. Calculate the imaginary coefficient only.

    $$-4-3=-7$$

15. Put these two results into a + bi form.

    $$5-7\times i$$

16. Both results are equal.

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

Answer

The final answer is shown below.