1. **Problem statement:** Express the complex numbers $2 + 3i$ and $1 - 2i$ in polar form, then use De Moivre's theorem to evaluate their product $(2 + 3i)(1 - 2i)$. Express the result in the form $a + ib$ and in exponential form. 2. **Polar form of a complex number:** A complex number $z = x + yi$ can be expressed in polar form as $$z = r(\cos \theta + i \sin \theta)$$ where $r = \sqrt{x^2 + y^2}$ is the modulus and $\theta = \tan^{-1}(\frac{y}{x})$ is the argument. 3. **Find polar form of $2 + 3i$:** - Modulus: $$r_1 = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$$ - Argument: $$\theta_1 = \tan^{-1}\left(\frac{3}{2}\right)$$ So, $$2 + 3i = \sqrt{13}(\cos \theta_1 + i \sin \theta_1)$$ 4. **Find polar form of $1 - 2i$:** - Modulus: $$r_2 = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}$$ - Argument: $$\theta_2 = \tan^{-1}\left(\frac{-2}{1}\right) = \tan^{-1}(-2)$$ So, $$1 - 2i = \sqrt{5}(\cos \theta_2 + i \sin \theta_2)$$ 5. **Product using polar form:** $$(2 + 3i)(1 - 2i) = r_1 r_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$$ Calculate modulus of product: $$r = \sqrt{13} \times \sqrt{5} = \sqrt{65}$$ Calculate argument of product: $$\theta = \theta_1 + \theta_2 = \tan^{-1}\left(\frac{3}{2}\right) + \tan^{-1}(-2)$$ 6. **Evaluate $\theta$ numerically:** - $\tan^{-1}(\frac{3}{2}) \approx 0.9828$ radians - $\tan^{-1}(-2) \approx -1.1071$ radians So, $$\theta \approx 0.9828 - 1.1071 = -0.1243$$ radians 7. **Express product in polar form:** $$\sqrt{65}(\cos(-0.1243) + i \sin(-0.1243))$$ 8. **Convert to rectangular form:** $$a = \sqrt{65} \cos(-0.1243) \approx 8.0623 \times 0.9923 = 7.997$$ $$b = \sqrt{65} \sin(-0.1243) \approx 8.0623 \times (-0.1240) = -1.000$$ So, $$(2 + 3i)(1 - 2i) \approx 7.997 - 1.000i$$ 9. **Express in exponential form:** $$re^{i\theta} = \sqrt{65} e^{-0.1243 i}$$ **Final answers:** - Rectangular form: $$7.997 - 1.000i$$ - Exponential form: $$\sqrt{65} e^{-0.1243 i}$$