Mathematics 10 — Exercise Review · Q6 (iii)

Full question

For z₁ = 5 + 4i, z₂ = 3 + 2i, evaluate z̅₁z̅₂.

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.

   $$4\times i\times \left(2\times i+3\right)+5\times \left(2\times i+3\right)$$

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

   $$5\times 3+5\times 2\times i+4\times 3\times i+4\times 2\times ii$$

3. Real × real: multiply the numerical coefficients.

   $$5\times 3=15$$

4. Real × imaginary: multiply the numerical coefficients.

   $$5\times 2\times i=10\times i$$

5. Imaginary × real: multiply the numerical coefficients.

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

6. Imaginary × imaginary: multiply the numerical coefficients.

   $$4\times 2\times ii=8\times i^2$$

7. Substitute these four products into the expansion.

   $$15+10\times i+12\times i+8\times i^2$$

8. In the last term replace i² by −1.

   $$8\times i^2=8\times \left(-1\right)$$

9. Multiply its coefficient by −1.

   $$8\times \left(-1\right)=-8$$

10. Rewrite the expansion after this sign calculation.

    $$15+10\times i+12\times i-8$$

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

    $$15-8+i\times \left(10+12\right)$$

12. Calculate the sum of the real terms.

    $$15-8=7$$

13. Calculate the sum of the imaginary coefficients.

    $$10+12=22$$

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

    $$22\times i+7$$

15. Take the conjugate of the product.

    $$7-22\times i$$

Answer

The final answer is shown below.