1. **State the problem:** Given complex numbers $z_1 = 2 + i$, $z_2 = 3 + 4i$, and $z_3 = \overline{z_1}$ (the conjugate of $z_1$), express the following in the form $a + bi$: a) $z_2 z_3$ b) $\frac{z_2}{z_3}$ 2. **Recall the conjugate:** The conjugate of $z_1 = a + bi$ is $\overline{z_1} = a - bi$. So here, $z_3 = 2 - i$. --- ### a) Calculate $z_2 z_3$ 3. Use the formula for multiplication of complex numbers: $$ (a + bi)(c + di) = (ac - bd) + (ad + bc)i $$ 4. Substitute $z_2 = 3 + 4i$ and $z_3 = 2 - i$: $$ (3 + 4i)(2 - i) = (3 \times 2 - 4 \times (-1)) + (3 \times (-1) + 4 \times 2)i $$ 5. Calculate each part: $$ = (6 + 4) + (-3 + 8)i = 10 + 5i $$ --- ### b) Calculate $\frac{z_2}{z_3}$ 6. To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator: $$ \frac{z_2}{z_3} = \frac{3 + 4i}{2 - i} \times \frac{2 + i}{2 + i} = \frac{(3 + 4i)(2 + i)}{(2 - i)(2 + i)} $$ 7. Calculate numerator: $$ (3 + 4i)(2 + i) = (3 \times 2 - 4 \times (-1)) + (3 \times 1 + 4 \times 2)i = (6 - 4) + (3 + 8)i = 2 + 11i $$ 8. Calculate denominator: $$ (2 - i)(2 + i) = 2^2 - (i)^2 = 4 - (-1) = 5 $$ 9. So the division is: $$ \frac{2 + 11i}{5} = \frac{2}{5} + \frac{11}{5}i $$ 10. Show cancellation step: $$ \frac{\cancel{2 + 11i}}{\cancel{5}} = \frac{2}{5} + \frac{11}{5}i $$ --- **Final answers:** a) $z_2 z_3 = 10 + 5i$ b) $\frac{z_2}{z_3} = \frac{2}{5} + \frac{11}{5}i$