1. **State the problem:** We have complex numbers $z_1 = 2 + i$, $z_2 = 3 + 4i$, and $z_3 = \overline{z_1} = 2 - i$ (the conjugate of $z_1$). We need to express the following in the form $a + bi$: a) $z_2 z_3$ b) $\frac{z_2}{z_3}$ c) $\frac{|z_2|}{|z_1|}$ 2. **Recall formulas and rules:** - Multiplication of complex numbers: $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$ - Division of complex numbers: $\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2 + d^2}$ - Modulus of complex number: $|a+bi| = \sqrt{a^2 + b^2}$ --- ### a) Calculate $z_2 z_3$ 3. Multiply $z_2 = 3 + 4i$ and $z_3 = 2 - i$: $$z_2 z_3 = (3 + 4i)(2 - i)$$ 4. Expand using distributive property: $$= 3 \times 2 + 3 \times (-i) + 4i \times 2 + 4i \times (-i)$$ $$= 6 - 3i + 8i - 4i^2$$ 5. Simplify terms and recall $i^2 = -1$: $$= 6 + 5i - 4(-1) = 6 + 5i + 4 = 10 + 5i$$ **Answer a:** $10 + 5i$ --- ### b) Calculate $\frac{z_2}{z_3}$ 6. Use the division formula: $$\frac{z_2}{z_3} = \frac{3 + 4i}{2 - i} = \frac{(3 + 4i)(2 + i)}{(2)^2 + (-1)^2}$$ 7. Calculate denominator: $$2^2 + (-1)^2 = 4 + 1 = 5$$ 8. Multiply numerator: $$(3 + 4i)(2 + i) = 3 \times 2 + 3 \times i + 4i \times 2 + 4i \times i = 6 + 3i + 8i + 4i^2$$ 9. Simplify numerator: $$= 6 + 11i + 4(-1) = 6 + 11i - 4 = 2 + 11i$$ 10. Write fraction: $$\frac{2 + 11i}{5} = \frac{2}{5} + \frac{11}{5}i$$ **Answer b:** $\frac{2}{5} + \frac{11}{5}i$ --- ### c) Calculate $\frac{|z_2|}{|z_1|}$ 11. Calculate modulus of $z_2 = 3 + 4i$: $$|z_2| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 12. Calculate modulus of $z_1 = 2 + i$: $$|z_1| = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$$ 13. Compute ratio: $$\frac{|z_2|}{|z_1|} = \frac{5}{\sqrt{5}}$$ 14. Simplify by rationalizing denominator: $$\frac{5}{\sqrt{5}} = \frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5}}{5} = \sqrt{5}$$ **Answer c:** $\sqrt{5}$